Thesis Front Matter - University of Calgary

UCGE Reports

Number 20208

Department of Geomatics Engineering

Improving Tracking Performance of PLL in High Dynamic

Applications

(URL: http://www.geomatics.ucalgary.ca/links/GradTheses.html)

by

Ping Lian

November 2004

THE UNIVERSITY OF CALGARY

Improving Tracking Performance of PLL in High Dynamic Applications

by

Ping Lian

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF GEOMATICS ENGINEERING

CALGARY, ALBERTA

NOVEMBER, 2004

© Ping Lian 2004

iii

ABSTRACT

The Phase-locked loop (PLL) is used in GPS receivers to track an incoming signal and to

provide accurate carrier phase measurements. However, the PLL tracking performance

and measurement accuracy are affected by a number of factors, such as signal-to-noise

power ratio, Doppler frequency shift, the GPS receiver�s jitter caused by vibration, and

the Allan deviation. Among these factors, the thermal noise and Doppler shift are the

most predominant and have a large influence on the design of the PLL. In high dynamic

situations, the conflict between improving PLL tracking performance and the ability to

track the signal necessitates some compromises in PLL design. This thesis investigates

the strategies to resolve this conflict.

Three methods are investigated to improve PLL tracking performance in high dynamic

applications: a Kalman filter-based tracking algorithm, application of a wavelet de-

noising technique in PLL, and an adaptive bandwidth algorithm. The Kalman filter-based

tracking algorithm makes use of a carrier phase dynamic model and a measurement from

the output of the discriminator to estimate the phase difference between the incoming

signal and the Numerical Controlled Oscillator (NCO) output, Doppler frequency and the

change rate of Doppler frequency. The wavelet de-noising technique effectively

decreases the noise level and allows broadening of the PLL bandwidth to track high

dynamics signals. The adaptive bandwidth PLL algorithm adapts the bandwidth of the

PLL according to the estimation of the incoming signal dynamics and noise level.

The performance is evaluated in terms of signal-to-noise ratios and dynamic variations

using simulating signals. The first two methods are found to produce better

improvements when the signal-to-noise ratio is low and the signal dynamic is high. The

iv

third method works well under high signal-to-noise ratios and less random dynamic

variations.

The results show that these methods outperform the ordinary PLL under high dynamic

conditions and the resulting carrier phase measurement is more accurate.

v

ACKNOWLEDGEMENTS

I would like thank my supervisor Dr. Gérard Lachapelle for his support and guidance

throughout my graduate studies. His continuous inspiration and advice were greatly

appreciated. The lessons I learned from his conscientious attitude and commitment to

graduate students will benefit me in future work.

I would also like to thank many graduate students and research associates. Particular

thanks go to Dr.Changlin Ma and Olivier Julien for their invaluable help. Many thanks

must go to Dr. Mark G. Petovello, Sameet Deshpande, Syed Salman, Zhi Jiang,

Dharshaka Karunanayake, Tao Hu, MinMin Lin and Haitao Zhang for the many

discussions and advice.

Finally thanks are given to my family, to my husband for his support and understanding,

to my son who gives me the motivation to study and live in a new country and my parents

for their encouragement.

vi

TABLE OF CONTENTS Approval Page..................................................................................................................... ii Abstract .............................................................................................................................. iii Acknowledgements..............................................................................................................v Table of Contents............................................................................................................... vi List of Tables ................................................................................................................... viii List of Figures .................................................................................................................... ix List of Abbreviations ....................................................................................................... xiii Chapter 1 : INTRODUCTION.............................................................................................1

1.1 Background.................................................................................................1 1.2 Objectives and Contributions .....................................................................5 1.3 Outline ........................................................................................................6

Chapter 2 : PHASE -LOCKED LOOP REVIEW ...............................................................8 2.1 Phase-Locked Loop ....................................................................................8

2.1.1 Basic Principle of the Phase-Locked Loop ......................................8 2.1.2 Loop Filter ......................................................................................11 2.1.3 Voltage-Controlled Oscillator and Numerical or Digital Controlled Oscillator .................................................................................................14 2.1.4 Phase Model and Key Parameters of a PLL...................................18 2.1.5 PLL Responses to Different Excitation Signals .............................20 2.1.6 Noise in the PLL.............................................................................24

2.2 PLL Tracking Loop Measurement Errors.................................................27 2.2.1 Thermal Noise ................................................................................28 2.2.2 Dynamic Stress ...............................................................................28 2.2.3 Vibration.........................................................................................29 2.2.4 Allan Deviation ..............................................................................30 2.2.5 Total PLL Tracking Loop Measurements Errors and Thresholds..30

2.3 COSTAS Loop..........................................................................................31 Chapter 3 : GPS SOFTWARE RECEIVER REVIEW......................................................35

3.1 GPS Software Receiver Structure.............................................................35 3.1.1 Signal Acquisition ..........................................................................36 3.1.2 Signal Tracking ..............................................................................37 3.1.3 Navigation Solution........................................................................39

3.2 Software PLL (SPLL)...............................................................................40 Chapter 4 : KALMAN FILTER BASED TRACHING ALGORITHM............................43

4.1 Kalman Filter Review...............................................................................43 4.1.1 Kalman Filter Algorithm ................................................................43 4.1.2 Adaptive Kalman Filter Algorithm ................................................46

4.2 Design Scheme .........................................................................................49 4.3 Kalman Filter-Based Tracking Algorithm................................................52

4.3.1 System Model .................................................................................52 4.3.2 Measurement Model .......................................................................55

Chapter 5 : APPLYING WAVELET DE-NOISING TECHNIQUE IN PLL ...................57 5.1 Wavelet De-noising Review.....................................................................57

5.1.1 Introduction ....................................................................................57

vii

5.1.2 Wavelet Transform.........................................................................58 5.1.3 Signal Decomposition ....................................................................61 5.1.4 Signal Reconstruction.....................................................................64

5.2 Wavelet De-noising by Soft-thresholding ................................................65 5.2.1 Decomposition................................................................................65 5.2.2 Threshold Detail Coefficients.........................................................66 5.2.3 Reconstruction................................................................................68

5.3 Applying Wavelet De-noising Technique in PLL....................................69 Chapter 6 : ADAPTIVE BANDWIDTH ALGORITHM ..................................................72

6.1 Design Scheme and PLL Linear Model....................................................72 6.2 Estimation of Signal Dynamics ................................................................75 6.3 Optimal bandwidth ...................................................................................80

Chapter 7 : TEST RESULTS AND ANALYSIS ..............................................................83 7.1 Test Configuration ....................................................................................83

7.1.1 Test Scheme....................................................................................83 7.1.2 GPS Receiver Configuration ..........................................................85

7.2 Horizontal Motion Testing, Results and Analysis....................................87 7.2.1 Real Time Processing .....................................................................91 7.2.2 Post Processing.............................................................................106

7.3 Three Dimensional Motion Test, Results andAnalysis ..........................111 7.3.1 Real Time Processing ...................................................................114 7.3.2 Post Processing.............................................................................126

7.4 Summary.................................................................................................130 Chapter 8 : CONCLUSIONS AND RECOMMENDATIONS .......................................137

8.1 Conclusions.............................................................................................137 8.2 Recommendations...................................................................................139

References 141 Appendix A: Derivation of the steady-state error............................................................146

viii

LIST OF TABLES Table 4.1: Allan Variance Parameters for Various Clocks .......................................... 54 Table 7. 1: Noise Characteristics ................................................................................... 85 Table 7. 2: Signal Tap Configuration............................................................................. 86 Table 7. 3: PLL Configuration ....................................................................................... 87 Table 7. 4: Real-Time Statistical Results for the Horizontal Motion .......................... 131 Table 7. 5: Real-Time Statistic Results for the Three Dimensional Motion................ 132 Table 7. 6 : Post Processed Statistic Results for the Horizontal Motion....................... 133 Table 7. 7: Post Processed Statistic Results for the Three Dimensional Motion......... 134 Table 7. 8: Real-Time Doppler Frequency Improvement............................................ 135 Table 7. 9: Post-Processing Doppler Frequency Improvement ................................... 136

ix

LIST OF FIGURES Figure 2.1: A Typical PLL Block Diagram................................................................... 9 Figure 2.2: Block diagram of a first order loop filter .................................................. 12 Figure 2.3: Block diagram of a second order loop filter ............................................. 13 Figure 2.4: Block Diagram of a NCO ......................................................................... 15 Figure 2.5: Carry Function of the Phase Accumulator................................................ 16 Figure 2.6: Linear phase model of a digital PLL......................................................... 19 Figure 2.7: Block Diagram of COSTAS Loop............................................................ 32 Figure 3.1: Generic GPS software receiver block diagram......................................... 36 Figure 3.2: Signal tracking block diagram .................................................................. 38 Figure 3.3: PLL Algorithm Implementation in Software............................................ 42 Figure 4.1: Adaptive Kalman Filter Algorithm (R unknown)..................................... 48 Figure 4.2: Adaptive Kalman Filter Algorithm (Q and R unknown) .......................... 49 Figure 4.3: Design scheme for using a Kalman filter in a PLL................................... 50 Figure 5.3: Signal Decomposition............................................................................... 64 Figure 5.4: Signal Reconstruction ............................................................................... 64 Figure 5.5: Decomposition of the signal ..................................................................... 66 Figure 5.7: Hard and Soft Thresholding...................................................................... 68 Figure 5.7: Reconstruction of the signal...................................................................... 69 Figure 5.8: Applying Wavelet De-noising Technique in a PLL ................................. 70 Figure 6.1: Adaptive Bandwidth Algorithm Design Scheme...................................... 73 Figure 6.2: Linear phase model of a digital PLL......................................................... 74 Figure 7.1: Test Configuration .................................................................................... 84 Figure 7.2: The Block Diagram of GPS Front End..................................................... 86 Figure 7.3 Simulated Horizontal Vehicle Trajectory ................................................. 88 Figure 7.4 Doppler Frequency of Satellite 9 .............................................................. 88 Figure 7.5: Doppler frequency from an ordinary PLL for satellite 9 (PLL Bandwidth =

18 Hz, C/NO = 45 dB-Hz) ......................................................................... 89 Figure 7.6: Doppler frequency from an ordinary PLL for satellite 9 (PLL Bandwidth =

18 Hz, C/NO = 39 dB-Hz) ......................................................................... 90 Figure 7.7: Doppler frequency from an ordinary PLL for satellite 9 (PLL Bandwidth =

0 Hz, C/NO = 39 dB-Hz) .......................................................................... 90 Figure 7.8: Doppler frequency from Kalman filter-based tracking algorithm for

satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz) ......................... 93 Figure 7.9: Doppler frequency from Kalman filter-based tracking algorithm for

satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz) ......................... 93 Figure 7.10: Doppler frequency from Kalman filter-based tracking algorithm for

satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) ......................... 94 Figure 7.11: PLL lock indicator for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 . 95 dB-Hz)....................................................................................................... 95 Figure 7.12: The PLL lock indicator for satellite 9 (PLL Bandwidth = 18 Hz, C/NO =

39 dB-Hz).................................................................................................. 96 Figure 7.13: The PLL lock indicator for satellite 9 (PLL Bandwidth = 30 Hz, C/NO =

39 dB-Hz).................................................................................................. 96

x

Figure 7.14: Doppler frequency after applying wavelet de-noising technique in the PLL for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz).................... 97



Figure 7.17: The PLL lock indicator for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz).................................................................................................. 99

Figure 7.18: The PLL lock indicator for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz).................................................................................................. 99

Figure 7.19: The PLL lock indicator for satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz)................................................................................................ 100

Figure 7.20: True acceleration from the simulator for satellite 9................................ 102 Figure 7.21: Estimated acceleration using the adaptive bandwidth algorithm for satellite

9............................................................................................................... 102 Figure 7.22: True jerk from the simulator for satellite 9............................................. 103 Figure 7.23: Estimated jerk using the adaptive bandwidth algorithm for satellite 9... 103 Figure 7.24: Doppler frequency after adapting the bandwidth for satellite 9 (C/NO is 45

dB-Hz)..................................................................................................... 105 Figure 7.25: Adaptive bandwidth for satellite 9.......................................................... 105 Figure 7.26: PLL lock indicator for satellite 9 ............................................................ 106 Figure 7.27: Doppler frequency from Kalman filter based tracking algorithm for

satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz) ....................... 107 Figure 7.28: Doppler frequency from Kalman filter-based tracking algorithm for

satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz) ....................... 107 Figure 7.29: Doppler frequency from Kalman filter-based tracking algorithm for

satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) ....................... 108 Figure 7.30: Doppler frequency after applying wavelet de-noising technique for



satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) ....................... 110 Figure 7.33: Doppler frequency after adapting the PLL bandwidth for satellite 9 (C/NO

is 45 dB-Hz)............................................................................................ 111 Figure 7.34: Three Dimensional trajectory 360º turns and 3.8 g lateral acceleration in

horizontal plane and 5 m/s velocity in height direction.......................... 111 Figure 7.35: Horizontal trajectory with 360º turns and 3.8 g lateral acceleration....... 112 Figure 7.36: Doppler frequency from the simulator for satellite 9.............................. 112 Figure 7.37: Doppler frequency from an ordinary PLL for satellite 9 (PLL Bandwidth =

18 Hz, C/NO = 45 dB-Hz) ....................................................................... 113 Figure 7.38: Doppler frequency from an ordinary the PLL for satellite 9 (PLL

Bandwidth = 18 Hz, C/NO = 39 dB-Hz) ................................................. 113 Figure 7.39: Doppler frequency from an ordinary the PLL for satellite 9 (PLL

Bandwidth = 30 Hz, C/NO = 39 dB-Hz) ................................................. 114

xi

Figure 7.40: Doppler frequency from the Kalman filter-based tracking algorithm for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz) ....................... 116






Figure 7.46: Doppler frequency after applying wavelet de-noising technique in PLL for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz) ....................... 119

Figure 7.47: Doppler frequency after applying wavelet de-noising technique in PLL for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz).................. 120

Figure 7.48: Doppler frequency after applying wavelet de-noising technique in PLL for satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) ....................... 120




Figure 7.52: The true acceleration from the simulator for satellite 9.......................... 123 Figure 7.53: The estimated acceleration using the adaptive bandwidth algorithm for

satellite 9 ................................................................................................. 123 Figure 7.54: The true jerk from the simulator for satellite 9 ....................................... 124 Figure 7.55: The estimated acceleration using the adaptive bandwidth algorithm for

satellite 9 ................................................................................................. 124 Figure 7.56: The Doppler frequency after adapting the bandwidth for satellite 9 (C/NO

is 45 dB-Hz)............................................................................................ 125 Figure 7.57: Adaptive bandwidth for satellite 9.......................................................... 125 Figure7. 58: Doppler frequency from Kalman filter based tracking loop for satellite 9

(PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)........................................ 126 Figure7. 59: Doppler frequency from Kalman filter based tracking loop for satellite 9

(PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz)........................................ 127 Figure 7.60: Doppler frequency from Kalman filter based tracking loop for satellite 9

(PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz)........................................ 127 Figure 7.61: Doppler frequency after applying wavelet de-noising technique for



satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) ....................... 129

xii

Figure 7.64 : Doppler frequency from adaptive bandwidth algorithm for satellite 9 (C/NO is 45 dB-Hz) ................................................................................. 130

xiii

LIST OF ABBREVIATIONS

GPS Global Positioning System DLL Delay Lock Loop PLL Phase-Locked Loop PLAN Position, Location and Navigation S/N Signal-to-Noise Ratio FLL Frequency-Locked Loop NCO Numerical-Controlled Oscillator LPLL Linear Phase-Locked Loop DPLL Digital Phase-Locked Loop ADPLL All-digital Phase-Locked Loop SPLL Software Phase-Locked Loop PD Phase Detector LF Loop Filter VCO Voltage-Controlled Oscillator DCO Digital-Controlled Oscillator SNR Signal-to-Noise Ratio LNA Low Noise Amplifier FPGA Field Programmable Gate Arrays C/A Coarse Acquisition BPSK Binary Phase Shift Keying TTFF Time to First Fix RMS Root Mean Square I Inphase Q Quadrature DSP Digital Signal Processing DWT Discrete Wavelet Transform STFT Short-Time Fourier Transform CWT Continuous Wavelet Transform IF Intermediate Frequency C/NO Carrier-to-Noise Ration dB deciBel

dBm deciBel one Milliwatt

1

CHAPTER ONE: INTRODUCTION

Research and development continue to expand the capabilities and to increase the

robustness of GPS receivers. Due to their flexibility, software receivers are quite valuable

and convenient in evaluating these potential improvements. For GPS receivers, the

technique used to compute the user�s position is based on the pseudorange and carrier

phase measurements. The common method of pseudorange and carrier phase

measurement is to use a Delay-Locked Loop (DLL) for code phase measurements and a

Phase-Locked loop (PLL) for carrier phase measurements. Unfortunately, a PLL can not

meet precision requirements all the time especially under high dynamic situations and

weak signals. Many methods have been developed to address this issue. Compared to

existing hardware, software receivers have more flexibility and advantages in improving

the tracking performance under high dynamic situations. This thesis investigates three

methods of PLL design in the software receiver to improve PLL tracking performance for

high dynamic applications.

1.1 Background

In practice, a GPS receiver must create the PRN code and carrier frequency plus Doppler

frequency using a DLL and PLL to track the incoming signals by synchronizing its local

carrier and code with the incoming signals. The accuracy of the frequency and phase

synchronization depends on the parameters of the DLL/PLL, the characteristics of the

incoming signals (such as signal-to-noise ratio and Doppler frequency), and the receiver

clock quality. In a high dynamics situation, the Doppler frequency changes rapidly with

2

time. This gives rise to a dilemma in GPS receiver design on the pre-integration and

PLL bandwidth. To tolerate dynamic stress, the most effective way is to broaden the PLL

bandwidth and reduce the pre-integration time. However, in order to decrease the thermal

noise and improve the tracking performance, a narrow PLL bandwidth and longer pre-

integration time are required. In reality, some compromise must be made to resolve this

conflict, especially under a high dynamic stress situation.

A software GPS receiver provides maximum flexibility in the design. It allows for the

design scheme to be easily simulated and implemented. It also meets the cost-effective

requirement for upgrading the system easily with the development of the new technology

� re-configurability. This research takes the advantage of software receiver technology

and tests the proposed algorithms in a GPS software receiver developed by the �PLAN�

group in our department, namely GNSS_SoftRx� (Ma et al 2004)

The common method for designing a PLL tracking loop is to choose the loop bandwidth

which is mainly determined by the loop filter considering the worst case of S/N and the

highest Doppler frequency caused by the dynamics. Usually these designs are robust but

not optimal. During a period of low dynamics, the loop bandwidth is not optimal for

reducing the tracking errors, as the signal-to-noise ratio is inversely proportional to the

loop bandwidth.

FLL (Frequency-Locked Loop) assisted PLL is another widely used method due to its

easy implementation (Jovancevic et al. 2003, Krumvieda et al. 2001). Under a low

3

dynamic situation, it can provide better performance due to the narrow bandwidth of

PLL. However, under high dynamic situations, it has to switch to FLL in order to offer

robustness. In this case the measurement accuracy deteriorates.

Open-Loop assisted Close-Loop PLL is also used in some applications (Yang 2003). This

method simply returns to the acquisition process when it operates under a high dynamics.

It can guarantee a reliable signal search in a wide spectral range. But the measurements

are very noisy therefore it is hard to obtain an accurate GPS carrier phase measurement

when it turns to the Open-Loop state.

Macabiau & Legrand (2000) tried to use �an extended Arctan discriminator� to increase

the PLL lose-lock threshold under high dynamics. Compared to an Arctan discriminator,

the extended Arctan discriminator has nearly a �π to + π range. They proposed use of the

extended Arctan discriminator in the L5 receiver�s pilot channel to benefit from the

absence of data on this channel. For L1/L2 receivers, it is restricted by the data bit

synchronization. If synchronization is not achieved, it cannot remove the effect of the

data bit transition.

A Doppler aiding carrier tracking loop provides the benefit of mitigating the phase error.

The basic concept of the Doppler aiding is to use internal or external Doppler information

to adjust the NCO frequency and therefore reduce, or cancel the effect of dynamic stress

(Demoz et al 2003). This allows narrowing of the bandwidth of the PLL even in high

4

dynamics. For closed carrier loop operation, the bandwidth of the loop is usually so

narrow that the aiding must be very precise with little or no latency.

The internal Doppler aiding is based on the estimation of the Doppler frequency inside

the GPS receiver. The most common example is to use the Doppler frequency derived

from PLL to aid the DLL through use of a scaling factor (Jovancevic et al. 2003).

External Doppler aiding is available from another sensor, i.e. an inertial navigation

system (Gold & Brown 2004). However, due to the errors in the sensor, the external

Doppler estimate is not always accurate. The disadvantage of this method is that the

quality of the sensor largely affects the Doppler estimate accuracy.

Adapting the PLL bandwidth based on the incoming signal dynamics is a better strategy

for improving the tracking performance of the PLL (Legrand & Macabiau 2000, Legrand

& Macabiau 2001). It was reported to have generated very good results, but no high

dynamics test results are presented. The work described herein implements the adaptive

bandwidth algorithm in a different way with special attention given to high dynamic

applications.

Although emphasis has been given to improving the tracking performance of PLL, the

reported accuracy is still not satisfactory specifically for high dynamic applications.

Accurate carrier phase measurement is one of the key issues for cm-level positioning

which has a broad range of applications. Therefore a thorough investigation and analysis

of improving PLL tracking performance are needed.

5

1.2 Objectives and Contributions

Given the deficiencies of previous PLL designs and the lack of research towards

satisfactory improvement of PLL tracking performance in high dynamic situations, this

thesis has the following objectives:

1. To develop algorithms to improve the tracking performance of PLL under

high signal dynamics;

2. To implement the proposed algorithms in the software receiver developed

by the �PLAN� group; and

3. To test and evaluate the performance of the proposed algorithms using a

simulator (STR6550) and SignalTap (front end of GPS receiver) produced

by the Accord company.

Based on the work described above, three algorithms are proposed that are implemented

in software. As mentioned earlier, the thermal noise and dynamic stress are the most

common errors and have a large influence on the design of PLL. This thesis focuses on

these two main error sources. Previous work did not achieve robustness and optimality at

the same time. The proposed algorithms try to improve the PLL tracking performance

with robustness and optimality in mind.

The proposed algorithms do not require external aiding and can be easily implemented in

a software receiver and therefore are cost-effective from this point of view. In addition,

they work in the real-time mode and can be used in many practical applications.

6

Another purpose of this research is to develop and test algorithms under a variety of

high dynamics situations and different signal-to-noise ratios. Previous research assumed

low or medium dynamics. This thesis complements the lack of results in this area and

presents many results based on simulated high dynamics signals. Comparisons between

the traditional design and the proposed designs are given.

1.3 Outline

In Chapter 2 of this thesis, there is a review of PLL theory followed by a detailed

explanation of the four main tracking errors and their effects on PLL tracking

performance. Following this is an introduction to the COSTAS loop which is widely used

in GPS receivers.

The existing GPS software receiver used for the investigations, namely GNSS_SoftRx�,

is introduced in Chapter 3, including its structure and a software PLL algorithm.

The proposed algorithms are presented in three Chapters. The Kalman filter-based

tracking algorithm is presented in Chapter 4. The Kalman filter algorithm is reviewed

first. A carrier phase dynamic model is developed based on the Kalman filter algorithm.

This is followed by a description on how to apply the model to the PLL.

Chapter 5 presents the method which applies the wavelet de-noising technique to PLL.

The basic wavelet theory is reviewed first. Details of the wavelet de-noising technique

7

are then presented with attention given to the method used in this thesis. The design

scheme combining the wavelet de-noising technique with PLL is introduced.

In Chapter 6, the Adaptive Bandwidth PLL Algorithm is presented. This involves the

algorithm methodology and the detailed implementation process.

The test results for the proposed algorithms presented in Chapter 4, 5, 6 are shown in

Chapter 7. These include a description of the testing configuration, the equipment used

and implementations tested.

Conclusions and recommendations are presented in Chapter 8.

8

CHAPTER TWO: PHASE -LOCKED LOOP REVIEW

GPS receivers track the Doppler frequency and phase to obtain very accurate carrier

phase measurements which are one of the important issues for many precise positioning

applications. Carrier phase tracking is accomplished using a PLL. This chapter reviews

PLL theory. Attention is given to PLL tracking capability with different types of exciting

signals and the related noise performance. Then the four main factors which affect PLL

tracking performance are explained in detail. Finally the Costas loop, which is widely

utilized in GPS receivers, is introduced.

2.1 Phase-Locked Loop

2.1.1 Basic Principle of Phase-Locked Loop

A PLL is a control loop which synchronizes its output signal (generated by a voltage or

numerical controlled oscillator) with a reference or input signal in frequency as well as in

phase. In the synchronized �often called locked � state the output frequency of a PLL is

exactly same as the input signal and the phase error between the oscillator�s output signal

and the reference signal is zero, or remains constant. In the unlocked state, the PLL

generates a control signal which is related to the phase error. This signal acts on the

oscillator in such a way that the phase error is again reduced to a minimum. In such a

control mechanism, the PLL always adjusts the phase of the output signal to lock to the

phase of the reference signal (Best 1999).

9

There are four different types of PLLs: LPLL (linear PLL), DPLL (�classical� digital

PLL), ADPLL (all-digital PLL) and SPLL (software PLL). The LPLL and SPLL are

relevant to this thesis.

A typical PLL block diagram is shown in Figure 2.1. It consists of three basic functional

components: a discriminator or a phase detector (PD), a loop filter (LF) and a voltage �

controlled oscillator (VCO).

Figure 2.1: A Typical PLL Block Diagram

Usually the discriminator is a multiplier and the loop filter is a low-pass filter. The output

of VCO maybe is a sine wave or a square wave. We assume here that the input signal is a

sine wave and the output signal is a square wave and it can be written as a Walsh function

(Best 1999) as follows

)](sin[)( 1111 ttUtu θω += (2.1)

)]([)( 2222 ttWUtu θω += (2.2)

where )(1 tu and )(2 tu are input and output signals, 1U and 2U are input and output

signals amplitudes, 1ω and 2ω are input and output signals frequencies, )(1 tθ , )(2 tθ are

input and output signals phases.

The Walsh function can be replaced by its Fourier series as follows

)(tud)(1 tu )(2 tu)(tucDiscriminator Loop filter VCO

10

+++= K))(3cos(

34))(cos(4)( 222222 ttttUtu θωπ

θωπ

(2.3)

For simplicity, we assume here that )(1 tθ is constant over time. The output of the

discriminator is given by

++++×+== K)3cos(4)cos(4)sin()()()( 2222112121 θω

πθω

πθω tttUUtututud (2.4)

When the LPLL is locked, we can obtain

12 ωω = (2.5)

+= Ked UUtu θπ

sin2)( 21 (2.6)

where 21 θθθ −=e is the phase difference or phase error and is constant in the locked

state. The other terms in )(tud have high frequencies and will be filtered out by the loop

filter. Neglecting the high frequency terms we can obtain

edd Ktu θsin)( ≈ (2.7)

where π

212 UUKd = is called the discriminator gain. When the phase error is small,

)(tud can be expressed by a linear form as

edd Ktu θ≈)( (2.8)

When )(tud passes through the loop filter, the high frequencies are eliminated and the

direct current term and low frequency components will pass. For a VCO, its

instantaneous frequency )(toω is given by (Best 1999) as

)()( 2 tuKtw foo += ω (2.9)

11

where oK is called the VCO gain. The instantaneous phase )(toθ of VCO is the

integral of its instantaneous frequency as

dttuKtdtttt

fot

oo )()()(020 ∫∫ +== ωωθ (2.10)

By comparing Equation 2.2 to Equation.2.10, we obtain

dttuKtt

fo )()(02 ∫=θ (2.11)

The control voltage from the loop filter adjusts the frequency and phase of the VCO to

synchronize with the input signal�s frequency and phase. In the locked state, the

frequency difference and the phase difference between the input signal and the output

signal from the VCO is zero. This means that the PLL replicates a signal whose

frequency and phase are the exact same as those of the input signal.

2.1.2 Loop Filter

The loop filter is a low-pass filter which passes through the control signals and filters out

most noise. The control signals are used to adjust the VCO frequency and phase to

provide accurate synchronization with the incoming signal. The loop filter order and

noise bandwidth determine the response of the loop to the input signal dynamics. The

orders of the commonly used loop filters are one or two. Figures 2.2 and 2.3 show the

block diagrams of the first and the second loop filters that are widely used in GPS

receivers (Kaplan 1996).

12

Figure 2.2: Block diagram of a first order loop filter (after Kaplan 1996)

Taking Figure 2.2 into account, the relationship between the input and output of the loop

filter is derived as follows

)()( 2 zTuzx doω= (2.12)

The transfer function from )(zx to )(zy is defined as (Hu 2001)

1

1

11

)()()( −

−

−+==

zz

zxzyzH xy (2.13)

Inserting Equation 2.12 into Equation 2.13 yields

)(11)(

11)( 2

1

1

1

1zTu

zzzx

zzzy doω−

−

−

−

−+=

−+= (2.14)

The output of the first order loop filter is described as

)()(21)( 2 zuazyzu dof ω+= (2.15)

Inserting Equation 2.14 into Equation 2.15 yields

)()(11

21)( 2

21

1zuazTu

zzzu dodof ωω +

−+= −

− (2.16)

Rearranging Equation 2.16 we obtain

uf(z) y(z)x(z)ud(z)

ωo2

a2ωo

T 1/2

Z-1

13

12

22

21 )()21()()

21()()1( −− −++=− zzuaTzuaTzuz doodoof ωωωω

(2.17)

Performing the inverse Z transform to Equation 2.17 yields

)1()21()()

21()1()( 2

22

2 −−+++−= nuaTnuaTnunu doodooff ωωωω (2.18)

Figure 2.3: Block diagram of a second order loop filter (after Kaplan 1996)

Based on the derivation for the first order loop filter, the relationship between the input

and output of the second order loop filter is easier to derive. From Equation 2.16 and

Figure 2.3, )(zx can be written as

)()(11

21)( 2

33

1

1zuazTu

zzzx dodo ωω +

−+= −

−

(2.19)

The output of the second order loop filter can be written as

)()(11

21)( 31

1zubzTx

zzzu dof ω+

−+= −

− (2.20)

ud(z)

uf(z)

a3ωo2

T 1/2

Z-1

T 1/2

Z-1

b3ωo

ωo3

x(z)

14

Inserting Equation 2.19 into 2.20

)()()(11

21

11

21)( 3

23

31

1

1

1zubzuazTu

zzT

zzzu dododof ωωω +

+

−+

−+= −

−

−

− (2.21)

Rearranging Equation 2.21 yields

)()24

(

)()22

()()24

()()21(

23

23

32

13

32

3

23

3221

zuzbTaT

zuzbTzubTaTzuzz

dooo

doo

dooo

f

−

−−−

+−+

−+++=+−

ωωω

ωωωωω

(2.22)

Performing the inverse Z transform of Equation 2.22, the output of the second order loop

filter is described as

)2()24

(

)1()22

()()24

()2()1(2)(

3

23

32

3

32

3

23

32

−+−+

−−++++−−−=

nubTaT

nubTnubTaTnununu

dooo

doo

dooo

fff

ωωω

ωωωωω

(2.23)

In general, the loop filter is a linear time-invariant system and its discrete form can be

described by the following difference equation (Hu 2001)

∑ ∑−

=

−

=

−+−−=1

1

1

0)()()(

N

k

N

rdrfkf rnubknuanu (2.24)

where N is the order of the loop.

2.1.3 Voltage-Controlled Oscillator and Numerical or Digital Controlled Oscillator

In a LPLL, one of the three components is a Voltage-Controlled Oscillator (VCO). One

important characteristics of a VCO is that it is an oscillator. What makes it different from

an ordinary oscillator is that its oscillating frequency is controlled by an external control

signal-voltage or sometime current. In an ideal situation, the oscillating frequency of the

15

VCO changes proportionally with the control voltage and the relationship was

described by Equation 2.9.

In a software PLL, the VCO is in a digital form and usually called a NCO (Numerical-

Controlled Oscillator) or DCO (Digital-Controlled Oscillator). The block diagram of a

NCO is shown in Figure 2.4.

Figure 2.4: Block Diagram of a NCO (Analog devices 2004)

A NCO can be implemented from a phase accumulator which consists of an adder, a

holding register and a Lookup table which is stored in a Read-Only-Memory (ROM). In

this case, the digital amplitude information that corresponds to a completed cycle of a

sine-wave or cosine-wave is stored in the ROM. Every amplitude value corresponds to a

certain phase value. As the sine or cosine wave is periodic, the whole signal wave over a

certain period time can be formed by reading out from the Look-up table periodically.

Every time the clock outputs a pulse, the phase value in the phase accumulator adds a

phase step once. This phase value is then used to address the look-up table to look for the

corresponding amplitude. The output frequency of a NCO is determined by the phase

step, sometime called a frequency selecting word. A higher frequency corresponds to a

sine or cosine

Clock fs

Frequency selecting word M

Phase accumulator

Adder Holding register Lookup table

16

larger frequency selecting word. The period character of the sine or cosine wave is

implemented by means of the carry function of the Phase accumulator which functions as

a �phase wheel� (Analog devices 2004) (see Figure 2.5). Each designated point on the

phase wheel corresponds to the equivalent point on a cycle of the sine-wave. As the phase

accumulator operates the vector rotates around the wheel, the corresponding output sine-

wave is read out from the look-up table. One revolution of the vector around the phase

wheel at a constant speed, results in one complete cycle of the output sine-wave. The

number of the discrete phase points contained in the wheel is determined by the

resolution of the phase accumulator. For a K bit length phase accumulator, this number

is K2 .

Figure 2.5: Carry Function of the Phase Accumulator (from Analog devices 2004)

A phase-to-amplitude lookup table is used to convert the instantaneous output value

(sometimes a truncated value) of the phase accumulator into the sine-wave amplitude

information that is the output of the NCO. The amplitude information of the lowest

frequency Ksf

2 is stored in ROM. The output frequency of the NCO is determined by the

17

jump size M which corresponds to the phase increment for each clock period. The

higher the output frequency is, the larger the jump size and the smaller the number of

discrete points during one cycle. According to the sampling theory, the highest output

frequency of the NCO is determined by the clock frequency. It is equal to half of the

clock frequency. The lowest output frequency equals to Ksf

2. In order to output the lowest

frequency, all amplitude information stored in ROM is read out cyclically. By contrast,

with the lowest frequency, the highest frequency only reads out twice every cycle

(Analog devices 2004).

Many NCOs exploit the symmetrical nature of a sine-wave or cosine-wave and utilizes

mapping logic to synthesize a complete sine-wave cycle to save the space on ROM. In

this case, only ¼ of the cycle data is required to be stored in the look-up table. The phase-

to-amplitude lookup table generates all the necessary data by reading forward, then back

through the lookup table.

The output frequency of the NCO is given by (Analog devices 2004)

Ks

oMff2

=

(2.25)

where M is called the frequency selecting word, K is the length of the phase

accumulator, sf is the clock frequency and of is the output frequency of the NCO. The

resolution of the output frequency is Ksf

2.

18

2.1.4 Phase Model and Key Parameters of a PLL

During the tracking period, the phase difference is zero or small. The PLL can be

regarded as a linear system. In order to analyze the tracking performance of the PLL, a

linear phase model is required. For the discriminator, it is linear when the phase

difference is small and is described by Equation 2.8. Besides, all loop filters are linear

time-invariant systems and their transfer function in the Z domain is described as

(Legrand & Macabiau 2001)

11

1

0

)1()( −−

−

=

−

−=∑

N

N

n

nn

z

zbzF (2.26)

The output phase of the NCO in the discrete domain can be derived from Equation 2.11

∑−

=

−+−==1

022 )1()1()()(

n

kfofo nuKnkuKn θθ (2.27)

Performing the Z transform for Equation 2.27, we can derive the model of NCO in the Z

domain as

1

12

1)()(

)( −

−

−==

zzK

zUz

zH o

fN

θ (2.28)

The simplified linear phase model of a digital PLL is shown in Figure 2.6

)(zU f

)(zeθ

)(2 zθ-

+

)(1 zθ )( zF

1

1

1 −

−

− zzK o

dK

19

Figure 2.6: Linear phase model of a digital PLL

The transfer function of the loop is defined by (Best 1999) as follows

11

1

1

2

)()1()(

)()()( −−

−

+−==

zzKFzzzKF

zzzH

θθ (2.29)

Inserting Equation 2.26 into Equation 2.29 we obtain

∑

∑−

=

−−−

−

=

−−

+−= 1

0

11

1

0

1

)1()( N

n

nn

N

N

n

nn

zbKz

zbKzH (2.30)

where do KKK = is called as the loop gain

The number of the poles in the transfer function is defined as the order of the loop.

The error transfer function of the loop is defined by (Best 1999) as follows

(2.31)

A set of key parameters which govern the dynamic performance of the PLL (Best 1999)

are as follows:

• The lock range Lw∆ . This is the frequency range within which a PLL locks within one

single-beat note between the reference frequency and the output frequency. Normally

the operating-frequency range of a PLL is restricted to the lock range.

• The pull-out range POw∆ . This is the dynamic limit for stable operation of a PLL. If

tracking is lost within this range, a PLL normally will lock again, but this process can

be slow if it is a pull-in process.

∑−

=

−−−

−

+−

−== 1

0

11

1

1 )1(

)1()()()( N

n

nn

N

Nde

e

zbKz

zKzzzH

θθ

20

• The pull-in range Pw∆ . This is the range within which a PLL will always become

locked, but the process can be rather slow.

• The hold range Hw∆ . This is the frequency in which a PLL can statically maintain

phase tracking. A PLL is conditionally stable only within this range.

In many designs, the relationship between these parameters is set by the following

inequality:

HPPOL wwww ∆<∆<∆<∆ . (2.32)

2.1.5 PLL Responses to Different Excitation Signals

Let us assume that the PLL is locked at the initial moment. The tracking performance of

the PLL responds to three important excitation signals which are calculated and analyzed

with attention given to the steady-state response. The excitation signals are phase step,

frequency step and frequency ramp.

2.1.5.1 A Phase Step Excitation Signal

If there is a phase step in the input signal at time 0=t , the digitized input phase )(1 nθ is a

step function and can be given by

( ) φθ ∆= )(1 nun (2.33)

where )(nu is a unit step function and φ∆ is the size of the phase step. The Z transform

of )(1 nθ is given by

21

φθ ∆−

=1

)(1 zzz

(2.34)

From Equation 2.31, the phase difference is given by

)()1(

)1()()()( 11

0

11

1

1 zzbKKz

zKzzHz N

n

nndo

N

Nd

ee θθθ∑

−

=

−−−

−

+−

−== (2.35)

Inserting Equation 2.34 into Equation 2.35 and using the Final Value Theorem, the

steady-state phase difference is given by

φθθ ∆−+−

−−=−=

∑−

=

−−−

−

→→∞ 1)1(

)1()1(lim)()1(lim 1

0

11

1

11 zz

zbKKz

zKzzz N

n

nndo

N

Nd

zeze (2.36)

Analyzing Equation 2.36, it is found that

10 ≥=∞ Nforeθ (2.37)

Equation 2.37 leads to the conclusion that any loop can track a phase step excitation

signal with no phase difference.

2.1.5.2 A Frequency Step Excitation Signal

If there is a frequency step ω∆ in the input signal, the digitized input phase )(1 nθ is given

by

( ) )(1 nnun ωθ ∆= (2.38)

Performing a Z transform to Equation 2.38, we obtain

21 )1()(

−∆=

zzz ωθ (2.39)

22

Similarly, using the Final Value Theorem the steady-state phase difference is given by

21

0

11

1

11 )1()1(

)1()1(lim)()1(lim

−∆

+−

−−=−=

∑−

=

−−−

−

→→∞ zz

zbKKz

zKzzz N

n

nndo

N

Nd

zeze ωθθ (2.40)

Analyzing Equation 2.40 (derivation is in Appendix A), the following conclusion is

obtained

≥

=∆=∞

20

10

N

NbKoe

ωθ (2.41)

Equation 2.41 means that the first order loop (no loop filter) can track a frequency step

excitation signal but with a constant phase difference and other loops whose orders are

larger than one can track a frequency step excitation signal with no phase difference.

2.1.5.3 A Frequency Ramp Excitation Signal

If there is a frequency ramp tω∆ in the input signal, the digitized input phase )(1 nθ is

given by

)(21)( 2

1 nunn ωθ ∆= (2.42)

Performing the Z transform on Equation 2.42 we obtain

31 )1()1(

2)(

−+∆=

zzzz ωθ (2.43)

Again, using the Final Value Theorem, the steady-state phase difference is given by

31

0

11

1

11 )1()1(

2)1(

)1()1(lim)()1(lim

−+∆

+−

−−=−=

∑−

=

−−−

−

→→∞ zzz

zbKKz

zKzzz N

n

nndo

N

Nd

zezeωθθ (2.44)

23

Analyzing Equation 2.44 (derivation is in Appendix A), the following conclusion is

obtained

>

=+

∆=∞

=∞

20

2)(

1

10N

NbbK

N

oe

ωθ (2.45)

Equation 2.45 means that the first order loop (no loop filter) cannot track a frequency

ramp excitation signal; the second order loop can track but with a constant phase

difference and other loops whose orders are larger than two can track a frequency ramp

excitation signal with no phase difference.

Summarizing the above discussion of three excitation signals, the conclusion is that the

Nth order loop can track the Nth order variation in the input phase with a constant phase

difference and can track less than the Nth order variation in the input phase with no phase

difference but can not track higher than the Nth order variation in the input phase.

It is obvious from the above derivation that the higher order variations is in the input

signal, the higher order of the loop is needed to track it.

In addition, the following three conditions are necessary for a PLL system to maintain

phase tracking (Best 1999)

• The frequency variation in the input signal must be within the hold range.

• The maximum tolerable frequency step must be smaller than the pull-out range.

24

• The rate of change of the reference frequency is limited by the loop natural

frequency.

Note that the discussion in this section is based on the assumption that the initial state of

the PLL is locked. If the PLL is unlocked initially, the above analysis is not suitable due

to the uncertain phase error. In most cases, the phase error is so large that it is

unreasonable to use the linear model. Instead of the tracking process, an acquisition

process with a nonlinear model can be used to analyze the PLL response.

2.1.6 Noise in the PLL

It is hard to derive the exact solutions for noise performance due to its randomness.

However, some simulated and experimental results can be used to discuss how the noise

affects the PLL. We assume that all noise signals discussed here are white.

Supposing that white noise with power nP is superimposed on the input signal of the PLL

with power sP , the bandwidth of the noise spectrum is limited to Bi by a pre-filter. If the

input signal is a sine wave, with the effect of noise, the zero crossings of the resulting

signal vary back or forward depending on the instantaneous polarity of the noise signal.

This so-called phase jitter or phase noise is designated by )(1 tnθ (Best 1999). The signal-

to-noise ratio at the input of the PLL is defined as

( )n

Si P

PSNR = (2.46)

25

The square of the RMS phase noise is given by (Best 1999)

iS

nn SNRP

P)(2

12

21 ==θ (2.47)

That is, the square of the RMS value of the phase jitter is inversely proportional to the

SNR at the input of the PLL. The square of the spectral density of the phase jitter

)(1 jwnθ can be expressed as (Best 1999)

2)(

212

1i

nn B

jwθθ = (2.48)

where iB is the bandwidth of the pre-filter

For a linear system, the spectral density of the output signal is equal to the product of the

spectral density of the input signal with the transfer function of the system. Based on this

property, the spectral density of the phase noise at the output can be calculated using

)()()( 21

22 ωθωωθ jjHj nn = (2.49)

Integrating Equation 2.48 in the frequency domain, the RMS value of the phase noise at

the output of the PLL is obtained as (Best 1999)

( ) dffjHB

dffji

nnn ∫∫

∞∞==

02

21

022

22 )2(

22 πθπθθ (2.50)

The integral dffjH∫∞

0

2)2( π is defined as the noise bandwidth BL (Best 1999)

dffjHBL ∫∞

=0

2)2( π (2.51)

For a second order loop, the solution of this integral is (Best 1999)

26

)41(

2 ζζω += n

LB

(2.52)

where ς is the damping factor and nw is the natural frequency.

Considering the transient response and noise performance, ζ = 0.7, BL = 0.53ωn are

chosen for most applications.

Substituting Equation 2.47 and 2.51 into 2.50, the output phase noise 22nθ can be rewritten

as (Best 1999)

i

L

s

nL

i

nn B

BPP

BB

×=×=2

212

2θθ (2.53)

By analogy to Equation 2.47, we can define a signal-to-noise ratio (SNR)L at the output

of the loop (Best 1999)

Ln SNR)(2

122 =θ (2.54)

Comparing Equation 2.53 and Equation 2.54, we can write (Best 1999)

L

ii

L

i

n

sL B

BSNRBB

PPSNR

2)(

2)( =×= (2.55)

For most applications, iB is much narrower than LB . Equation 2.55 shows that the PLL

has the advantage of improving the SNR of the input signal. The narrower the noise

bandwidth, the greater the improvement the PLL can provide.

27

One issue in the PLL design is how large iSNR)( must be to enable the safe

acquisition of the PLL. Experimental results (Best 1999) with second order loops have

demonstrated the following:

1. For )0(1)( dBSNR L = , a lock-in process will not occur because the output phase

noise is excessive;

2. At )3(2)( dBSNR L = , lock-in is eventually possible; and

3. For )6(4)( dBSNR L = , stable operation is generally possible.

For example, in the case of a second�order loop with MHzBi 2= and HzBL 18= ,

when dBSNR i 4.41)( −> , a stable operation is generally possible.

Another issue in the PLL design is how often on the average a PLL will temporarily

unlock. Defining avT as the average time interval between two lockouts, for second-order

loops, avT has been found to be a function of LSNR)( according to experimental results

and it gets longer as LSNR)( increases (Best 1999).

2.2 PLL Tracking Loop Measurement Errors

In precise positioning applications, carrier phase measurements are used to estimate

navigation related solutions. Therefore accurate Doppler frequency and phase

measurements play an important role in determining the positioning precision. The PLL

tracking loop measurement errors have a significant effect on the tracking performance.

This section discusses four dominant error sources that affect PLL phase errors: thermal

28

noise, dynamic stress, vibration and Allan deviation. The last two errors are related to

the clock used in the receiver.

2.2.1 Thermal Noise

Thermal noise is the most common present error source in a PLL. The PLL phase jitter

caused by thermal noise is computed as follows (Kaplan 1996):

+=

00 /211

/2360

NTCNCBn

t πσ (2.56)

where tσ is the 1-sigma thermal noise, nB is the carrier loop noise bandwidth (Hz),

ONC / is the carrier-to-noise ratio and T is the pre-integration time (s).

In order to mitigate the thermal noise error, a narrow bandwidth, high carrier over noise

ratios and longer pre-integration time are required.

2.2.2 Dynamic Stress

The dynamic stress error is closely related to the order of the loop. For a second order

loop, the dynamic stress error is defined as (Kaplan 1996)

2

2

2

2 2809.0n

e BdtdR

=θ (2.57)

where 2eθ is the 1-sigma dynamic stress error (deg), and 2

2

dtdR is the acceleration along

the line of sight (deg/sec2)

29

For a third order loop, the dynamic stress error is defined as (Kaplan 1996)

3

3

3

3 4828.0n

e BdtdR

=θ (2.58)

where 3

3

dtdR is the jerk along the line of sight (deg/sec3)

The dynamic stress is strictly dependent on signal dynamics and the bandwidth of the

loop for a given loop order. High dynamics produce high dynamic stress. However,

increasing the bandwidth will effectively decrease the dynamic stress error especially for

the higher order loop.

2.2.3 Vibration

Vibration induces clock phase noise which can be computed as follows (Kaplan 1996)

( )∫= max

min 22 )(

2360 f

f mm

mmv

Lv df

ffP

fSf

πσ (2.59)

where vσ is the 1-sigma vibration induced clock phase noise, Lf is the L-band input

frequency (Hz), )(2mv fS is the oscillator vibration sensitivity of Lff /∆ per G as a

function of mf , mf is the random vibration modulation frequency (Hz) and )( mfP is the

power curve of the random vibration as a function of mf ( HzG /2 ).

Vibration induced clock phase noise has no relationship with the loop order and the

bandwidth. It is mainly determined by the vibration environment.

30

2.2.4 Allan Deviation

Allan deviation also induces clock phase noise. The equation for computing the Allan

deviation-induced phase noise for a second order PLL is (Kaplan 1996)

n

LAA B

f)(1442τσθ = (2.60)

where 2Aθ is the Allan deviation-induced jitter (deg), τ is the short-term stability gate

time for the Allan variance measurement (s) and )(τσ A is the Allan deviation.

The equation for computing short-term Allan deviation induced phase noise for a third

order PLL is (Kaplan 1996)

n

LAA B

f)(1603τσθ = (2.61)

Under the situation of a narrow bandwidth and poor clock quality, the Allan deviation

effect dominates the PLL tracking error. Clock quality is the key element in decreasing

this error (Kaplan 1996).

2.2.5 Total PLL Tracking Loop Measurements Errors and Thresholds

The 3-sigma value of the total PLL tracking loop measurement errors can be written as

(Kaplan, 1996)

eAvtPLL θθσσσ +++= 22233 (2.62)

where PLLσ3 is the 3-sigma total tracking error.

31

Equation 2.62 indicates that the dynamic stress error is a 3-sigma effect and is additive

to the phase jitter. According to the rule-of-thumb, the PLL tracking threshold is

computed as (Kaplan 1996)

o153

222 ≤+++ eAvt

θθσσ (2.63)

Equation 2.63 means that the 1-sigma value of the total errors must be less than 15

degrees. Otherwise the PLL lock state can not be guaranteed.

2.3 COSTAS Loop

One prominent use of the PLL is its carrier recovery or extraction from phase-coded or

phase shift keying modulated signals. Among these signals, the BPSK (binary phase shift

keying) modulated signal is the typical one which is acquired by changing the carrier

phase according to the modulating data bits. Data bit �0� corresponds to �180°� or �0°�

phase change in the carrier signal and �1� to �0°� or �180°� degree change. Recovering

the carrier frequency from this modulated signal is difficult because a randomly

modulated BPSK signal has no discrete energy line at the carrier frequency. In order to

extract the carrier frequency from the BPSK modulated signal, some special strategy is

needed to eliminate the effect of the data bit modulation. One well-known method is to

use a COSTAS loop.

Figure 2.7 shows a carrier-recovery loop for BPSK signals, called a COSTAS loop (Egan

1998). Both the output of the VCO and its 90° phase shift signal multiplies with the input

signal. The rationale of COSTAS is explained as follows.

32

Figure 2.7: Block Diagram of COSTAS Loop

Assuming the input BPSK signal is given by

( ) ( )( )ttUtu ii 11sin θϕω ++= (2.64)

where iU is the amplitude of the input signal and 1w and )(1 tθ are input carrier frequency

and phase and

==

=1mod00mod180

0

0

signalulatingwhensignalulatingwhenϕ

Assuming the output signal is given by

( ) ( )( )ttUtu oo 22sin θω += (2.65)

( ) ( )( )ttUtu oo 22' cos θω +=

where oU is the amplitude of the NCO output signal and 2w and )(2 tθ are frequency and

phase of the NCO output signal. (2.66)

The output of the first multiplier (neglecting the high frequency component) is

SI(t)

SQ(t)

uo(t)'

uo(t) ud(t) uf(t) ui(t)

Low-pass

Low-pass

VCO Loop filter

90° shift

33

( )( ))()(cos2

)()()( 2121 tttUUtututS oioiI θθϕωω −++−==

(2.67)

( )( ))()(sin2

)()()( 2121' tttUUtututS oi

oiQ θθϕωω −++−== (2.68)

At the output of the discriminator we obtain

( )( )))()((222sin8

)()()( 2121

22

tttUU

tStStu oiQId θθϕωω −++−== (2.69)

For BPSK signals, 2φ = 0 or 2π. Because the sine function�s period is 2π, the above

Equation can be rewritten as

( )( )))()((22sin8

)()()( 2121

22

tttUU

tStStu oiQId θθωω −+−== (2.70)

From the above derivation, it is obvious that after the discriminator, the effect of the

modulation has been removed. That is why a COSTAS loop is insensitive to a 180° phase

change in the input signals. Because of this advantage, a COSTAS loop is widely used to

recover the carrier frequency from the PSK signals.

Another advantage of the COSTAS loop is that it provides data detection simultaneously

with carrier recovery from the output of the low-pass filter. Under a locked state,

21 ωω = ,

0)()( 21 =− tt θθ ,

we can obtain

ϕcos2

)()()(

oi

oiI

UUtututS

=

= (2.71)

34

Assuming 2

oiI

UUU =

−

==0mod

1modcos)(

isdataulatingUisdataulatingU

UtSI

III ϕ (2.72)

From Equation 2.72, the modulated data can be detected by judging the output of the

low-pass filter )(tSI .

35

CHAPTER THREE: GPS SOFTWARE RECEIVER REVIEW

The tremendous increase of microprocessor computational power is leading to the

proliferation of software radio techniques in which signals are digitized and hence

processed entirely in the digital domain. The advantages offered by a software receiver

such as flexibility, reconfigurability and high fidelity make its application highly suitable

in the GPS field. In addition, GPS software receivers enable the verification of many

signal processing algorithms and a variety of advanced designs, thereby providing a very

convenient platform for a wide range of research, development and testing. In this

chapter, the GPS software receiver structure is presented first; then, the software PLL

algorithm used in this thesis is introduced.

3.1 GPS Software Receiver Structure

A typical GPS software receiver consists of the hardware front end and a signal

processing unit in software. The front end contains a LNA (low noise amplifier), two

stages of down convertors as well as a signal buffer. The digitized intermediate frequency

is fed into the signal processing unit which usually is implemented totally in software,

even though sometimes FPGA (field programmable gate arrays) is deployed to

accomplish the function of the correlators to speed up the calculation. A typical GPS

software receiver block diagram is shown in Figure 3.1.

36

Figure 3.1: Generic GPS software receiver block diagram

3.1.1 Signal Acquisition

Signal acquisition is the initial process, prior to signal processing. It consists of a two-

dimensional search in code phase and frequency. The receiver replicates the C/A code

and the rough Doppler frequency for each satellite. After Doppler frequency removal,

(and, strictly speaking, the carrier frequency thereof) a correlation process is

implemented that attempts to align the replica code with the one in the incoming signal.

At the conclusion of the signal acquisition part, the rough code phase and Doppler

frequency for each satellite should be known (Accord 2004).

Signal acquisition should be achieved in the shortest possible time for all visible

satellites. The almanac data, approximate time and the local position can aid the

acquisition process and significantly increase its speed. Fast acquisition reduces the time

to first fix (TTFF) and therefore improves the overall quality of the receiver.

Front End

Signal Acquisition

Signal Tracking

NavigationSolution

Signal Processing and Navigation Processing (Software)

37

3.1.2 Signal Tracking

The signal tracking process involves code wipe off and carrier wipe off. The code wipe

off utilizes a DLL (delay lock loop) to synthesize the replica code phase with the one of

the incoming signal. The carrier wipe off is accomplished by a carrier tracking loop to

synthesize the replica carrier frequency and phase with the one of the incoming signals

(Accord 2004).

The carrier tracking loop estimates the signal frequency and phase using an error

feedback mechanism. The common carrier tracking loop is the FLL which can only track

the signal frequency, or the PLL-Phase-locked loop which cannot only track the signal

frequency but also the signal phase and therefore provide an accurate carrier phase

measurement. A typical tracking loop block diagram is shown in Figure 3.2 (Ray 2003).

First, the digital IF is stripped off the carrier in the in-phase and quadra-phase mixers by

the replica carrier from the carrier loop NCO and generates the I and Q signals. The

amplitude of the signal can be determined from the vector sum of the I and Q components

and the phase angle can be determined from the arctangent of Q/I. The objective of the

carrier tracking loop is to track the phase of the incoming signal with no error. Otherwise

the carrier tracking loop will detect the phase difference between the incoming signal and

the replica signal and adjust the replica frequency and phase to reach alignment.

38

The carrier phase measurement is derived from the Doppler frequency and phase of the

NCO (Accord 2004).

Figure 3.2: Signal tracking block diagram (Accord 2

The I and Q signals are then correlated with early, prompt, and late rep

code generator. The phase shift between the early, prompt and late

usually 1/2 chips apart (In order to mitigate multipath effects, a sm

used). The prompt replica code tries to align with the incoming co

P

Doppler Removal

Doppler Removal

L

E

QE

Q

Q

IE

Pre-integration

Pre-integration

Pre-integration

Carrier Loop Filter

Pre-integration

Pre-integration

Pre-integration

Digital IF Input

Carrier Loop NCO

S

CodFilteDis

Code NCO

CodeGen.

IL

IP

Carrier Loop Discr.

e r&cr.

90° hifter

004)

lica codes from the

replica codes are

aller phase shift is

de. The maximum

P

L

39

correlation is obtained if alignment is achieved. In this case, the early and late

correlators will half the maximum correlation. Any misalignment will produce a

difference in the magnitudes of the early and late correlators which can be detected and

corrected by the code tracking loop (DLL). The phase and adjusted chips at the alignment

state are employed to calculate the pseudorange measurement.

Following the correlators are pre-integration units that increase the signal power.

However, the pre-integration time is determined based on the signal power and user

dynamics, etc. When the signal is weak, a longer pre-integration period is required to

accumulate high signal power. Nevertheless, high signal dynamics applications present

an opposite requirement. A compromised design is necessary in order to meet these two

requirements simultaneously.

3.1.3 Navigation Solution

A navigation algorithm combines raw measurements from the signal processing with the

GPS satellite orbit data to estimate position related parameters. Fundamentally, a

navigation solution is an estimate of the user�s position and any other required

parameters. A GPS estimator is used to estimate the required parameters. The typical

estimated states of a GPS receiver are three position components, the receiver clock

offset and the clock drift. The velocity is often added in dynamic applications.

40

3.2 Software PLL (SPLL)

With advancements in microprocessors and digital signal processing, innovative software

radio technology emerges. Nowadays, a lot of hardware may be replaced by software

methods, providing increased flexibility and ease of reconfiguration that make software

more competitive. Along with this trend, implementation of the PLL in software is used

in many systems. As long as the speed of the microprocessors or the DSP (digital signal

processing) is fast enough, the PLL algorithms can be realized in real time. Even though

all types of PLL can be implemented in software, this thesis will focus on an LPLL-like

software implementation due to its straightforward realization. Based on the knowledge

of the underlying PLL components in digital forms, the PLL algorithm is developed in

software as shown in Figure 3.3. The software receiver used to test the LPLL

implementations studied herein is GNSS_SoftRx� developed by the PLAN group (Ma

et al 2004).

In the algorithms implemented herein, the first step is to initialize the NCO. The next step

is the implementation of the discriminator and loop filter functions. The control signal

from the output of the loop filter changes the frequency of the NCO. According to the

frequency change, the phase step is adjusted and consequently the phase index is updated

to output the signal with a new frequency. The last step is to save the current values of

the outputs of the discriminator and the loop filter which will be used for the next

iteration.

41

)()()( 21 nununu d =

∑ ∑−

=

−

=−+−−=

1

1

1

0)()()(

N

k

N

rdrfkf rnubknuanu

)(00 nuK f+= ωω

s

K

fM

πω2

2⋅=

M = integer of M

)2()1(

)()1()2()1(

)()1(

+−=+−

=−+−=+−

=−

NnuNnu

nunuNnuNnu

nunu

ff

ff

dd

dd

M

M

( ) ( )( )[ ]nindexphasetablenu

Mnindexphasenindexphase_sin_)(

1__

2 =+−=

Create a sin-table of NCO

Kstepphase22_ π= (N: the length of phase accumulator)

For n =1 to 2K

[ ] ( )( )0_

sinsin__)1()(

==

+−=

indexphasenphasentable

stepphasenphasenphase

( )nu1

42

Figure 3.3: PLL Algorithm Implementation in Software

43

CHAPTER FOUR: KALMAN FILTER BASED TRACKING ALGORITHM

This chapter first briefly reviews the well-known Kalman filter algorithm. It includes two

adaptive Kalman filter algorithms. Then the design scheme using the Kalman filter in the

tracking loop is presented. This is followed by a detailed discussion about how to

perform a Kalman filter-based tracking algorithm in the PLL.

4.1 Kalman Filter Review

4.1.1 Kalman Filter Algorithm

The Kalman filter is essentially a recursive algorithm that implements a predictor-

corrector type estimator. The predictor is based on a system model and the corrector is

based on the measurement model. The Kalman filter is optimal in the sense that it

minimizes the estimated error covariance. Let us consider any linear system which can be

described using the following state model

)()()( tButFxtx +=& (4.1)

where x is the state vector, F is the dynamic matrix, B is input matrix and u is the

vector of input white noise. The estimation is based on the measurements which can be

modeled as

)()()( tvtHxtz += (4.2)

44

where z is the measurement vector, H is the measurement matrix and v is the vector of

the measurement noise.

For a discrete Kalman filter, the system model and measurement model can be expressed

as (Salychev 1998)

11,11, −−−− +Φ= kkkkkkk wGxx (4.3)

kkkk vxHz += (4.4)

where 1, −Φ kk is called the transition matrix and kw and kv are white noise sequences

with the following properties:

0)()( == kk vEwE (4.5)

=

=other

lkifQwwE T

lk 0)( (4.6)

=

=other

lkifRvvE T

lk 0)( (4.7)

The estimation of x is derived using the system model and the measurement model from

the condition (Salychev 1998):

==−− )~~(])�)(�[( Tkk

Tkkkk xxEtracexxxxEtrace minimum (4.8)

where kx� is the estimate of kx and kx~ is the estimation error. The solution of the above

Equation is described by the algorithm of the discrete Kalman filter which is well known.

Here, only the final results are given (Salychev 1998):

)�(�� 11,11, −−−− Φ−+Φ= kkkkkkkkkk xHzKxx (4.9)

where kK is the Kalman filter gain which is calculated by (Salychev 1998)

45

11,1, ][ −

−− += RHPHHPK Tkkkk

Tkkkk

(4.10)

where 1, −kkP is a priori covariance matrix of the estimation error and kP is the covariance

matrix of the estimation error and defined as )~~( Tkkk xxEP = . kP is caused by the modeling

errors and it provides an indication of the accuracy of the estimate. They can be

expressed as (Salychev 1998)

Tkkkk

Tkkkkkkk QGGPP 1,1,1,11,1, −−−−−− +ΦΦ= (4.11)

1,)( −−= kkkkk PHKIP (4.12)

Equation 4.9 consists of two terms. The first one is the solution of the deterministic

system model. It estimates x using the system model neglecting the noise effects. The

second is the correction to the estimation based on the measurements. The difference

between the current measurement and a priori estimate is called the innovation

sequence kν and is given by (Salychev 1998)

kkkk

kkkkkk

kkkk

kkkkkk

vxHxHvxH

xHzxHz

+=

−+=

−=

Φ−=

−

−

−

−−

1,

1,

1,

11,

~�

��v

(4.13)

where 1,� −kkx is a priori estimation and 1,~

−kkx is a priori estimation error.

It is obvious from the above equation that the innovation sequence includes the

information of the prior estimation error which is derived from the measurements but is

corrupted by the measurement noise. In order to reduce the difference between the

46

estimate and the measurement, the estimated error is added to the estimated vector

weighted by the Kalman filter gain. If the measurement is noisy, the Kalman filter gain

will be low, and not too much correction will be done to the estimate. In contrast, an

estimated error derived from the accurate measurement is weighted largely to correct the

estimation.

In practice, the system model can never be exactly correct. Therefore the Kalman filter

cannot estimate the state vector with absolute accuracy. The innovation sequence plays

an important role in the Kalman filter algorithm. Especially in the adaptive algorithm, it

is chosen as an indicator of the estimation error.

4.1.2 Adaptive Kalman Filter Algorithm

From the above discussion, it is clear that the Kalman filter algorithm estimation

accuracy depends on the knowledge of the system model and noise statistics. An

inaccurate system model and noise statistics deteriorate the performance of the Kalman

filter. In reality, sometimes it is hard to obtain accurate noise characteristics. In these

cases, an adaptive Kalman filter algorithm provides a better solution.

Here we deal with two typical cases: R is unknown and R and Q are unknown. Let�s

deal with the first case where R is unknown. From Equation 4.13, we can derive

kT

kkkkTkkk RHPHEC +== −1,)vv( (4.14)

If kC is known, R can be derived from Equation 4.14 as

47

Tkkkkkk HPHCR 1, −−=

(4.15)

For a stationary system, kC can be estimated using (Salychev 1998)

Ti

k

iik k

C vv1�1∑=

= (4.16)

or

Tkkkk k

Ck

kC vv1�1�1 +−= − (4.17)

For a non-stationary system, kC can be estimated using (Salychev 1998)

TkkkC vv� = (4.18)

Summarizing the above derivation, the scheme for adaptive Kalman filter algorithm when

Q is unknown is shown in Figure 4.1.

If Q and R are unknown, we need to derive Q and R . From Equations 4.3, 4.9 and 4.13,

we can write

11,11,�� −−−− =Φ− kkkkkkk WGxx (4.19)

kkkkkk Kxx v�� 11, =Φ− −− (4.20)

Comparing Equation 4.19 with Equation 4.20, we obtain

kkkkk KWG v�11, =−− (4.21)

Calculating the covariance of the two sides of the above equation, the result is

Tk

Tkkk

Tkkkkk KEKGQG )vv(�

1,11, =−−− (4.22)

Under the condition 1−≈ kk KK , Equation 4.22 can be written as

48

Tkkk

Tkkkkk KCKGQG 111,11,

��−−−−− =

(4.23)

The scheme for an adaptive Kalman filter algorithm when Q and R are unknown is

shown in Figure 4.2 (Salychev 1998).

Figure 4.1: Adaptive Kalman Filter Algorithm (R unknown)

kk Px ,�

kz

11,1, �� −−− Φ= kkkkk xx

Tkkkk

Tkkkkkkk QGGPP 1,1,1,11,1, −−−−−− +ΦΦ=

Tkkkkkk HPHCR 1,

��−−=

1,)( −−= kkkkk PHKIP

11,1, ][ −

−− += kT

kkkkT

kkkk RHPHHPK

1,�v −−= kkkkk xHz

Tkkk

Ti

k

iik Cor

kC vv�vv1�

1== ∑

=

kkkkkk Kxx v�� 11, +Φ= −−

49

Figure 4.2: Adaptive Kalman Filter Algorithm (Q and R unknown)

4.2 Design Scheme

kk Px ,�

kz

11, �� −− Φ= kkk xx

Tkkkk

Tkkkkkkk QGGPP 1,1,1,11,1, −−−−−− +ΦΦ=

Tkkkkkk HPHCR 1,

��−−=

1,)( −−= kkkkk PHKIP

11,1, ]�[ −

−− += kT

kkkkT

kkkk RHPHHPK

=−−T

kkkk QGG 1,1,Tkkk KCK 11

�−−

1,�v −−= kkkk xHz

Tkkk

Ti

k

iik Cor

kC vv�vv1�

1== ∑

=

kkkkkk Kxx v�� 11, +Φ= −−

50

The design scheme for using a Kalman filter in carrier tracking loop is shown in Figure

4.3

Figure 4.3: Design scheme for using a Kalman filter in a PLL

If the Kalman filter is removed, Figure 4.3 is a common COSTAS loop with two pre-

integration filters. The pre-integration filter performs integration and dump functions.

The integration time is determined by the incoming signal dynamics and signal-to-noise

ratio. In order to increase the signal power, longer integration time is needed. However,

in high dynamics situations, Doppler frequency changes faster with time and a longer

pre-integration time cannot provide an accurate Doppler frequency during the pre-

uo(n)

uI(n)

ui(n)

uQ(n)

uo(n)�

ud(n) uf(n)

Pre-integration filter

NCO

Loop filter

90° shift

Discriminator

Pre-integration filter

Kalman filter

Phase difference Doppler frequency Change rate of Doppler frequency

SQ(n)

SI(n)

51

integration interval. Consequently, the pre-integration effect deteriorates and in this

case a short pre-integration time is preferred. In this thesis, 1 ms is used.

The output from the discriminator is very noisy and fed into the loop filter. Noise that is

beyond the bandwidth of the loop is filtered out by the loop filter. Noise which is within

the bandwidth of the loop will pass through the loop filter, go into the NCO and will

subsequently affect the tracking performance. Especially for high dynamic applications

which require wide bandwidths, more noise will pass through the loop filter and cause the

carrier phase measurement to be very noisy. In this thesis, the PLL bandwidth is

configured at a high value of 18 Hz in order to track the high signal dynamics. Under this

situation, the measurement from the PLL is very noisy and therefore the carrier

measurement is not satisfactory. To address this problem, this design uses a Kalman filter

to obtain less noisy measurements. Making use of this characteristic, the Kalman filter

can change the weight on the measurement according to the noise level of the

measurement. A much less noisy measurement can be obtained from a Kalman filter-

based tracking algorithm.

The Kalman filter-based tracking algorithm takes the discriminator output as a

measurement and deploys a carrier phase dynamic model to estimate the phase

difference, Doppler frequency and the change rate of the Doppler frequency. However,

the estimated phase difference and Doppler frequency are not used for Doppler removal.

They are only used as measurements. If the Kalman filter-based tracking algorithm is

used instead of the total PLL, the results often diverge. In order to keep reliable tracking

capability and at the same time provide more accurate measurements, this research

52

combines the PLL with the Kalman filter-based tracking algorithm. The PLL

guarantees a basic tracking ability and the Kalman filter-based tracking algorithm refines

the tracking performance with the help of the PLL tracking information (phase

difference).

4.3 Kalman Filter-Based Tracking Algorithm

The estimated parameters of the Kalman Filter are phase difference, Doppler frequency

and change rate of Doppler frequency. The measurement is the phase difference from the

output of the discriminator.

4.3.1 System Model

Assuming that the acceleration along the line-of-sight is constant, the Doppler frequency

in the incoming signal can be expressed as

tfftf avd +=)( (4.24)

where df is the overall Doppler frequency in the incoming signal, vf is the frequency shift

caused by the relative velocity and af is the change rate of the frequency shift caused by

the acceleration along the line-of-sight between the satellite and the receiver.

From Equation 4.24, the Doppler frequency over a period of t∆ is obtained as

tftfttf add ∆+=∆+ )()( (4.25)

53

The carrier phase variation caused by the Doppler frequency in the incoming signal

over a period of t∆ is

2

0

01

21

])([

)(

ttf

dftf

dtf

d

tad

td

∆+∆=

+=

+=∆

∫

∫∆

∆

ττ

ττθ

(4.26)

Now we can create the carrier phase dynamics model as (Psiaki 2000, Psiaki 2001, Psiaki

& Jung 2002, Jung & Psiaki 2003, Ziedan & Garrison 2003)

ndnco

ka

d

e

ka

d

e

Wft

fft

tt

ff

+

∆−

∆

∆∆

=

−100010001

00

10010

21

_

1

2

θθ (4.27)

where eθ is the carrier phase error between the incoming signal and the output of the

NCO, and dncof _ is the Doppler frequency replicated by the NCO at the moment k-1.

t∆ is the update period and equals to the pre-integration time. In this thesis, =∆t 1 ms is

chosen. [ ]Tadn WWWW θ= is a noise vector which consists of discrete Gaussian white

noise sequences .

The next task is to calculate the covariance of nW . Assuming that θW dW aW are

independent and the spectral intensity of the continuous-time white noise process of θW ,

dW and aW are ad QQQ ,,θ respectively, the corresponding spectral intensity matrix cQ of

continuous-time white noise process is given by

54

=

a

dc

QQ

QQ

000000θ

(4.28)

The covariance Q which corresponds to the discrete noise process can be calculated by

(Ziedan & Garrison 2003)

dtQQ Tkkckk

t

1,1,0 −−

∆ΦΦ= ∫ (4.29)

From Equation 4.27, we obtain

=Φ −1,kk

∆

∆∆

10010

21

2

t

tt

(4.30)

Substituting Equation.4.30 and 4.28 into Equation.4.29, we obtain

∆∆∆

∆∆∆

∆∆∆

+

∆∆

∆∆

+

∆=

ttt

ttt

ttt

Qtt

tt

Qt

QQ ad

26

238

6820

000

02

023

00000000

23

234

345

2

23

θ (4.31)

If we assume that θQ is caused by the receiver clock phase bias, dQ is caused by the

receiver clock frequency drift and aQ is caused by the acceleration between the satellite

and the receiver along the line-of-sight, θQ and dQ can be determined depending on the

receiver clock quality in the application and aQ can be determined depending on the

dynamics (Psiaki 2000, Psiaki 2001, Psiaki & Jung 2002, Jung & Psiaki 2003). θQ and

55

dQ can be calculated using the key Allan variance parameters 0h , 1−h and 2−h according

to the following formulas (Brown & Hwang 1992):

02hSQ f ==θ (4.31)

228 −== hSQ gd π (4.32)

where fS and gS are spectral amplitudes.

The typical Allan variance parameters for various clocks are listed in Table 4.1

(Brown & Hwang 1992). An ovenized crystal clock is used in this thesis,

Table 4.1: Allan variance parameters for various clocks

Clock types 0h 1−h 2−h

Crystal 19102 −× 21107 −× 20102 −×

Ovenized Crystal 20108 −× 21102 −× 23104 −×

Rubidium 20102 −× 24107 −× 29104 −×

4.3.2 Measurement Model

In this thesis, the output of the discriminator is chosen as the measurement. There are two

reasons for this selection. The first reason is just as mentioned before to simplify the

algorithm. If we can select a suitable discriminator which provides a linear relationship

with the carrier phase error, we can simply use an ordinary Kalman filter rather than more

complicated Extended Kalman filter. The second reason is to eliminate the data bits

56

effect. As discussed in Section 2.3, at the output of the discriminator the navigation

data is removed (the code is removed after correlation). Otherwise we have to consider its

effect in the Doppler frequency tracking. Here an Arctan discriminator is selected which

has a very good linear property within

−

2,

2ππ . The relationship between the inputs and

outputs of the discriminator is given by

)arctan(_I

Qmeae S

S=θ (4.33)

where meae _θ is the phase error between the incoming signal and the NCO. QS and IS are

outputs of the pre-integration filters.

The measurement can be modeled as avee _θ , which is the average of the phase error over

the period t∆ . avee _θ can be derived as follows:

tftkftkfk

dft

dkfkfkt

t

dncoade

dnco

t

ad

t

eavee

∆−∆−+∆−+−=

∆−−+−+−

∆=∆ ∫∫

∆∆

_2

_0

2

0_

21)1(

61)1(

21)1(

1])1(21)1()1([1)(

θ

τττττθθ (4.34)

The measurement model is given by (Psiaki 2000, Psiaki 2001, Psiaki & Jung 2002, Jung

& Psiaki 2003).

kdnco

ka

d

e

meae Vft

fftt +∆−

∆∆=

−

_

1

2

_ 2621

θθ

(4.35)

where kV is measurement noise which is a Gaussian white noise sequence and its

covariance matrix is R . In this thesis, an adaptive R algorithm described in section 4.12

is utilized.

57

CHAPTER FIVE: APPLYING WAVELET DE-NOISING TECHNIQUE IN PLL

This chapter starts with a review of some basic concepts of wavelet. The focus is on an

explanation of wavelet de-noising technique. This is followed by a detailed introduction

of how to implement the wavelet de-noising technique in a PLL.

5.1 Wavelet De-noising Review

5.1.1 Introduction

The Wavelet transform was introduced at the beginning of the 1980s. Since then, various

types of Wavelet transforms have been developed, along with many applications. The

continuous-time Wavelet transform has been found to be an effective tool for data

analysis. Compared to the continuous-time Wavelet transform, the discrete Wavelet

transform (DWT) is much more efficient and is suitable for much wider applications. The

DWT has an excellent compaction signal format and has therefore has the advantage of

being computationally very efficient. The DWT has been applied to many fields

including image compression, de-noising, numerical integration and pattern recognition

(Mertins 1999).

The most well known tool for signal analysts is Fourier analysis, which breaks down a

signal into constituent sinusoids of different frequencies. This means that Fourier analysis

58

transforms signals from the time domain to the frequency domain. However, Fourier

analysis has a serious drawback. In transforming to the frequency domain, time

information is lost. In the frequency domain, it is impossible to tell which frequencies the

signal contains at a given time. In order to overcome this deficiency, a �window�

technique was created which analyzes only a small section of the signal at a time. The

corresponding transform is called the Short-Time Fourier Transform (STFT). The STFT

maps a signal into a two-dimensional function of time and frequency. It provides the

signal�s frequency information at different periods of time. The drawback is that the

window size is fixed and is not flexible to analyze different frequencies. Many

applications require a more flexible approach; for example, for high frequencies it is

more efficient to deploy a narrow window and, for low frequencies, a wide window

(Mertins 1999).

Wavelet transform not only has the property of STFT but also has a distinctive one � a

variable-size window which uses a narrower time window to analyze high frequency

signals and a wider time window to analyze low frequency signals. Different from the

Fourier transform, Wavelet analysis uses a time-scale domain rather than time-frequency.

5.1.2 Wavelet Transform

A wavelet is a waveform of effectively limited duration that has an average value of zero

(The Mathworks, Inc. 2004). It is defined as

59

RbaRaa

btatba ∈≠∈

−= − ;0,,)( 2

1

, ψψ (5.1)

where )(tψ is called the mother wavelet or the basic wavelet which is the wavelet

prototype, a is the scale factor, b is a shift factor and R is the real domain. From Equation

5.1, it is obvious that the wavelet is acquired by scaling and shifting the basic wavelet.

Scaling a wavelet simply means dilating (or compressing) it. The larger (smaller) the

scale factor, the more �stretching� (�compressing�) is the wavelet. The scale factor is

related to the frequency of the signal. Shifting a wavelet simply means translating it in the

time domain (The Mathworks, Inc. 2004).

5.1.2.1 Continuous Wavelet Transform

Similar to the Fourier transform, the continuous wavelet transform (CWT) is defined as

the integration of the signal )(tf multiplied by the scaled, shifted versions of the wavelet

function ψ over time (The Mathworks, Inc. 2004) as

∫∫∞+

∞−

∗−∞+

∞−

−== dt

abtatfdtttfbaC ba ψψ 2

1

, )()()(),( (5.2)

where C is called a wavelet coefficient. This transform produces many wavelet

coefficients that are a function of scale and translation. C represents how much a specific

section of the signal correlates to a given wavelet. Higher C values correspond to high

similarity. In practice, C is calculated according to the following (The Mathworks, Inc.

2004): first choose a wavelet and compare it with the initial section of the signal to be

analyzed, then compute C using Equation 5.2, after that shift the wavelet from the initial

60

section to the end section of the signal and compute the corresponding C; and, lastly,

change the scale of the wavelet and repeat steps 2 and 3. This process is illustrated in

Figure 5.1.

Figure 5.1: Illustration of wavelet shift and scale concepts (from The Mathwork, Inc. 2004)

The wavelet transform produces a time-scale aspects of a signal. The scale and frequency

are related. A low scale corresponds to a compressed wavelet which shows the rapid

change feature of the signal. Consequently we obtain the high frequency character of the

signal. By contrast, a high scale corresponds to a stretched wavelet which shows the slow

change feature of the signal. Therefore we obtain the low frequency characteristic of the

signal.

5.1.2.2 Discrete Wavelet Transform (DWT)

The continuous wavelet coefficients have some redundancy. In some applications such as

signal coding or compression, time-scale factors are sampled so that the signal is

represented by wavelet coefficients in an economical manner. A typical choice is

zja j ∈= ,2 (5.3)

61

zkkb j ∈= ,2 (5.4)

where z is the integer domain.

First considering the case that the time is continuous but the scale and shift are sampled.

The wavelet in this case is written as

)2(2)( 2, kt jt

j

kj −= −−ψψ (5.5)

The corresponding wavelet transform is given by (The Mathworks, Inc. 2004)

∫∞+

∞−

−∗−−= dtktftC jt

j

ji )2()(2)( 2, ψ (5.6)

This produces a so-called �dyadic� grid in a time-scale domain as shown in figure 5.2

(The Mathworks, Inc. 2004).

Figure 5.2: Discrete wavelet analysis (from The Mathwork, Inc. 2004)

5.1.3 Signal Decomposition

An efficient way to calculate the wavelet coefficients was developed by Mallat (1989)

using filters. Based on this idea, a fast iterative algorithm was developed to compute the

coefficients efficiently. In this algorithm, a special wavelet prototype was constructed for

which the perfect reconstruction of the signal is possible. The synthesis takes the form

62

∑∑∈ ∈

=Zj Zk

kj tkjCtf )(),()( ,ψ . (5.7)

Equation 5.5 is an orthonormal basis of )(2 RL . Since a basis consists of linearly

independent functions, )(2 RL may be understood as the direct sum of subspaces (Mertins

1999) as

LL ⊕⊕⊕⊕= − 1012 )( WWWRL (5.8)

with

ZjZkkspanW jtj

j ∈

∈−= −−,),2(2 2ψ (5.9)

where jW represents a certain subspace which corresponds to a certain frequency band.

For a given j and Zk ∈ , ),( kjC gives the frequency information of the signal in the jth

sub-band. The signal component corresponds to this sub-band can be expressed as

jjZk

kjj WtcDtkjCtcD ∈=∑∈

)(,)(),()( ,ψ (5.10)

where )(tcDj is called the detail coefficient of the signal at level j. The signal can be

represented as

∑∈

=Zj

j tcDtf )()( (5.11)

Now we define the subspaces ZjV j ∈, as the direct sum of 1+jV and 1+jW (Mertins

1999)

11 ++ ⊕= jjj WVV (5.12)

and the subspace jV has the property

LL ⊂⊂⊂⊂ −+ 11 jjj VVV

63

For 1+jW , the lower j corresponds to the higher frequency components. Consequently

1+jV contains the low-pass components of jV . On the other hand, the subspace jV should

have the scaling property. For example )2( tf , which is obtained by scaling the signal

)(tf by the factor of two is in the subspace 1−jV and vice versa (Mertins 1999)

1)2()( −∈⇔∈ jj VtfVtf (5.13)

In addition, if we form a sequence of functions )(tcAj by projection of )()( 2 RLtf ∈ onto

the subspaces jV , based on Equation 5.12, we can derive the following expression

(Mertins 1999)

111 )(),()()( −−− ∈+= jjjjj VtcAtcDtcAtcA (5.14)

where )(tcAj is defined as the approximation of the signal at level j. Combining Equation

5.11 and Equation 5.14 the signal f can be expressed as

∑<

+=JJ

Jj cDcAf (5.15)

This means that the signal can be decomposed into several levels, and at each level, the

signal is decomposed into the approximation and the detail coefficients. The

approximation coefficients are the high-scale, low-frequency components of the signal.

The detail coefficients are the low-scale, high-frequency components.

The detail and approximation coefficients can be acquired by means of a set of low-pass

filters h and their complementary high-pass filters g. The decomposition algorithm starts

with signal f to calculate the approximation and detail coefficients at the first level, and

64

then those of the second level, and so on. This process is shown in Figure 5.3 (Mallat

1998, Zhang et al. 2001).

Figure 5.3: Signal Decomposition

5.1.4 Signal Reconstruction

Signal reconstruction is an inverse process of the signal decomposition. It begins at the

highest level and uses the detail and approximation coefficients to build the next lower

level approximation coefficient. The reconstruction process is illustrated in Figure 5.4

(Mallat 1998, Zhang et al 2001).

Figure 5.4: Signal Reconstruction

jcD

2−jcA1−jcA

1−jcDjcA 'h ↑2

↑2

'h

'g↑2

↑2

'g

1+jcA

2+jcD1+jcD

1+jcAjcA

h

↓2

↓2h

g

↓2

↓2

g

65

5.2 Wavelet De-noising by Soft-thresholding

The objective of de-noising is to suppress the noise part of a noisy signal in order to

recover the real signal.

Wavelet de-noising is one of the most significant applications of wavelets. The soft

thresholding-based de-noising algorithm has proven to have the advantages of optimizing

MSE (mean-squared error) and to keep the smoothness of the de-noised signal (Donoho

1995). Because of these advantages, the wavelet de-noising technique has been used in

this thesis.

The underlying model for the noisy signal is basically of the following form:

)()()( nenfns σ+= (5.16)

where )(nf is signal and )(ne is the noise with level σ . In the simplest model we

suppose that )(ne is a Gaussian white noise. The general de-noising procedure involves

three steps (Donoho 1995, Taswell 2000, Zhang et al 2001, Mertins 1999): first

decompose the signal using an octave-band filter bank implementing a discrete wavelet

transform, then threshold the wavelet coefficients to remove noise, and lastly reconstruct

the signal using the modified wavelet coefficients.

5.2.1 Decomposition

66

An effective decomposition algorithm is discussed in Section 5.1.3. Here what is

needed is to apply this algorithm to the noisy signal )(ns . First choose a wavelet, choose

a decomposition level N and then compute the wavelet decomposition (approximate

coefficient and detail coefficient) of the signal at each level. This process is illustrated in

Figure 5.5.

Figure 5.5: Decomposition of the signal

5.2.2 Threshold Detail Coefficients

In this step, for each level from 1 to N, a threshold is computed first and then a soft

thresholding is applied to the detail coefficients. The threshold is chosen based on

(Donoho 1995)

nnrthr )log(2

1σ= (5.17)

where 1r is a constant and can be set to 1 while choosing the orthogonal wavelet, and n is

the number of the data samples. Equation 5.17 shows that the threshold is proportional to

cD1 cA1

cA2 cD2

cAn cDn

cAn-

N

S

67

the noise level σ and also related to the length n of the noisy signal. Considering a

simple Gaussian white noise case, the detail coefficients at any level from the standard

DWT are Gaussian white noise with the same standard deviation as the noise levelσ .

Next the threshold is applied to the detail coefficients at each level. Two approaches were

proposed for this purpose: hard and soft thresholding. Both methods are illustrated in

Figure 5.6. The hard thresholding uses the following non-linearity (Mertins 1999):

≤>

=thrcDthrcDcD

mcDj

jjj 0

, (5.18)

where jmcD is called the modified detail coefficient.

The soft thresholding uses the following non-linearity (Donoho 1995):

+−= ))(sgn( thrcDcDmcD jjj (5.19)

where the sign �+� indicates a nonlinear process that only keeps positive values. It has the

following property

≤

>−=− +

thrcD

thrcDthrcDthrcD

j

jjj

0)(

The function sgn has the following property:

<−≥

=0101

)sgn(j

jj cDif

cDifcD (5.20)

Compared to hard thresholding, soft thresholding has several beneficial properties.

If thrcDJ ≤ , both the hard and soft thresholding do the same thing � set the detail

coefficients to zero. This process means that these detail coefficients are caused by noise

68

and should be cut off. If thrcDJ > , the hard thresholding assumes that they are signals

and do nothing about them. But soft thresholding assumes that noise is superimposed on

the signals and therefore it removes the noise from the signals, which acts like shrinking

the nonzero coefficients towards 0 by the amount of thr .Therefore, the soft-thresholding

provides a more effective de-noising result.

If the threshold is chosen properly, the noise level in the reconstructed signal is much

lower than the one in the original. If the threshold is too small, the improvement is not

very much. On the other hand, a threshold that is too large may cause signal distortion.

Therefore the de-noising performance depends on the proper selection of the threshold.

5000 5100 5200-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Detail Coefficient

Original detail coefficient

5000 5100 5200-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Detail Coefficient

Hard-thresholding detail coefficient

5000 5100 5200-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Detail Coefficient

Soft-thresholding detail coefficient

threshold

Figure 5.6: Hard and Soft Thresholding (from The Mathworks, Inc. 2004).

5.2.3 Reconstruction

69

Reconstruction is the inverse process of decomposition. Also the direction is inversed.

From the highest level N, reconstruction is performed using the original approximation

coefficients and the modified detail coefficients at level N. This process is shown in

Figure 5.7.

Figure 5.7: Reconstruction of the signal

From the above discussion, it is clear that the wavelet de-noising technique is different

from filtering. The former method decreases the noise part of the signal at any place of

the frequency bandwidth and at the same time keeps the frequency components of the

signal, whereas the latter method eliminates (ideally) the noise beyond the frequency

band of the signal but does nothing with the noise within the frequency bandwidth of the

signal.

5.3 Applying Wavelet De-noising Technique in PLL

cAn

cD1cA1

S

cA2 cD2

cDn

cAn-1

O

70

As discussed previously, there is a contradiction in the PLL design. Reducing noise

and improving the tracking performance require a narrow PLL bandwidth but tracking

high dynamic signals requires a wide one. Wavelet de-noising provides a better design

scheme to deal with this problem. The rationale is quite simple: First decrease the noise

level before the loop filter using a wavelet de-noising technique (the wavelet de-noising

technique can also be placed at the output of the loop filter). Because the noise level is

reduced, this allows one to increase the bandwidth of the loop to track high dynamic

signals. This design scheme is shown in Figure 5.8.

Figure 5.8: Applying Wavelet De-noising Technique in a PLL

The main function of the wavelet de-noising is to reduce the noise level within the

bandwidth of the loop filter. It is known that the noise beyond the bandwidth of the loop

filter will be filtered out by the loop filter, therefore only the noise within the bandwidth

of the loop filter will pass through and affect the NCO tracking performance. It should be

ui(n) uo(n)

SQ(n)

SI(n) uI(n)

uQ(n)

uo(n)�

ud(n) uf(n)

Pre-integration

NCO Loop filter

90° shift

Discriminator

Pre-integration

Wavelet de-noising

71

noted that the wavelet de-noising technique can only reduce the noise level rather than

eliminate the noise totally. The remaining noise still can affect the NCO tracking

performance but at a lower level.

The disadvantage of the wavelet de-noising technique is that it may cut off some useful

signals which are smaller than the threshold. This will make the PLL spend more time to

lock due to the loss of some control signals. However the decrease of the noise will

produce smaller phase error which will help the PLL to maintain locked. The results

shown in Chapter 7 indicate that the overall performance is improved.

72

CHAPTER SIX: ADAPTIVE BANDWIDTH ALGORITHM

The common solution of mitigating thermal noise and dynamic stress error is to choose

the noise bandwidth for the highest dynamics. This results in a sub-optimal use of the

loop during periods of low dynamics. The adaptive bandwidth algorithm is designed to

adapt the noise bandwidth of the PLL in real-time in order to achieve a better tracking

performance by estimating the signal dynamics. Aided by knowledge of the carrier-over-

noise power ratio from the software receiver, an optimal bandwidth is calculated. This

chapter first presents the design scheme. Then the method to estimate the incoming signal

dynamics is introduced. After that, the strategy for selecting the optimal bandwidth is

presented.

6.1 Design Scheme and PLL Linear Model

The design scheme of the adaptive bandwidth algorithm is shown in Figure 6.1

73

Figure 6.1: Adaptive Bandwidth Algorithm Design Scheme

As shown in Figure 6.1, this algorithm utilizes the phase difference from the output of the

discriminator to estimate the dynamics of the incoming signal and then adjusts the

parameters of the loop filter to change the PLL bandwidth. In order to obtain the

dynamics estimation, a digital PLL phase model is required.

Unlike LPLL, the analysis of the digital loop is based on Z the transform. The transfer

function of the loop filter in the Z domain is given by Equation 2.26 and the transfer

function of the NCO in the Z domain is given by Equation 2.28. The simplified linear

phase model of a digital PLL is shown in Figure 6.2

SQ(n)

SI(n) uI(n)

ui(n)

uQ(n)

uo(n)�

uo(n) ud(n) uf(n)

Pre-integration

NCO Loop filter

90° shift

Discriminator

Adaptive bandwidth Algorithm

Pre-integration

74

Figure 6.2: Linear phase model of a digital PLL

From this model, we can calculate the transfer function of the phase difference (Best

1999)

∑−

=

−−−

−

+−

−== 1

0

11

1

1 )1(

)1()()(

)( N

n

nn

N

Nde

e

zbKz

zKzzzH

θθ

(6.1)

where do KKK = .

When the PLL is determined, )(zHe is known. )(1 zθ is the input phase caused by the

Doppler frequency in the incoming signal. Therefore it is related to the incoming signal

dynamics. In order to estimate the signal dynamics, the expression of )(1 zθ should be

derived. Through the use of Equation 6.1, the relationship between the phase difference

and signal dynamics can be obtained. If the phase difference measurement is available,

the estimation of the signal dynamics is possible.

)(zU f

)(zeθ

)(2 zθ-

+

)(1 zθ )( zF

1

1

1 −

−

− zzK o

dK

75

6.2 Estimation of Signal Dynamics

The distance between the satellite and the GPS receiver along the line of sight can be

modeled as (Legrand & Macabiau 2000, Legrand & Macabiau 2001, Legrand &

Macabiau 2001)

!)(

0

)(0 m

tRtRm

m

m∑∞

=

= (6.2)

where R0(m) is the coefficient for each term. )1(

0R is the velocity of the second term, )2(0R

is the acceleration of the third term and )3(0R is the jerk of the fourth term. The relative

velocity along the line-of-sight is the derivative of the distance

)!1()()(

1

1

)(0 −

==−∞

=∑ m

tRtRdtdtv

m

m

m (6.3)

The relative velocity generates the Doppler frequency. The relationship between them is

given by

λ)()()( tvf

ctvtf d == (6.4)

where c is the velocity of light, f is the signal frequency (1575.42 MHz for L1

frequency) and the λ is the wavelength of the signal. Replacing Equation 6.3 in Equation

6.4, Equation 6.4 can be rewritten as

)!1(1)(

1

1

)(0 −

=−∞

=∑ m

tRtfm

m

md λ

(6.5)

It is known that phase is the integration of the frequency. Therefore the phase caused by

the Doppler frequency is written as

76

!2)(2)(

1

)(001 m

tRdttftm

m

mt

d ∑∫∞

=

==λππθ

(6.6)

Sampling )(1 tθ using a sample frequency sf (sample period is ST ), one can write

ss f

nnTt == (6.7)

In a discrete form, Equation 6.6 is written as

ms

m

m

m

fn

mR

n ∑∞

=

=1

)(0

1 !2)(λπθ . (6.8)

The sample number during each integration period is designated as IN which is equal to

the integration period multiplied by the sample frequency. Under a linear condition, the

synthesized input phase at the end of each integration period is the sum of the phase

during the integration period (Legrand & Macabiau 2000). Therefore we can obtain

)(!

2)(1

)1(1

)(0

1 ∑∑−

−=

∞

=

=I

I

kN

Nkn

m

mm

s

m

nfm

Rk

λπθ (6.9)

because (Legrand & Macabiau 2000)

mL

m

ms

mI

m

ms

kN

Nkn

m

Bk

fNk

f

nI

I ==∑

−

−=

1

)1( (6.10)

where I

sL N

fB = is the bandwidth of the pre-integration filter. In addition,

∑∑∞

=

∞

=

⋅==1

)(

1

)(0

1 !!2)(

0m

mm

mmL

mm

mk

BmkRk θ

λπθ (6.11)

where mL

mm

BR

λπ

θ)(

0)(0

2= (6.12)

77

Next we calculate the steady state of the phase difference that describes how the loop

tracks the input phase variation. The input phase in the Z domain is

][!

)(1

)(

10 m

m

m

kZm

z ∑∞

=

=θ

θ (6.13)

Based on Equation 6.1 the phase difference in the Z domain is obtained by

][!

)1(

)1()()()(

1

)(

1

0

11

1

10 m

m

m

N

n

nn

N

Nd

ee kZm

zbKz

zKzzHz ∑∑

∞

=−

=

−−−

−

+−

−==

θθθ . (6.14)

Using the Final Value Theorem of the Z transform, the steady-state phase difference can

be calculated by

[ ]

+−

−−=−=∞ ∑

∑

∞

=−

=

−−−

−

→→][

!)1(

)1()1(lim)()1(lim)(

1

)(

1

0

11

1

110 m

m

m

N

n

nn

N

Nd

zeze kZm

zbKz

zKzzz

θθθ . (6.15)

Through analysis (the detail analysis is in Appendix A), one concludes that for the terms

in the incoming signal whose orders are smaller than N , )(∞eθ equals zero, for the terms

whose orders are larger than N, )(∞eθ equals to infinite, and for the term whose order

equals to N, )(∞eθ is constant. This means that a Nth order loop can track the Nth order

variation in the input phase but with a constant steady-state phase difference. However, it

can track variations in the input phase whose order is less than N without error but cannot

track the variation whose order is larger than N. The conclusion is summarized in

Equation 6.16.

78

>∞

=

<

=∞∑

−

=Nm

NmbK

Nm

N

nnO

N

e 1

0

0

0

)(θ

θ

(6.16)

In this thesis, 1=OK .

Based on Equation 6.16, it can shown that the steady state phase difference for a second

order loop is

10

)2(0)(

bbe +=∞

θθ (6.17)

where 2

)2(0)2(

02

LBR

λπ

θ = . (6.18)

Similarly, the steady state phase difference for a third order loop is

210

)3(0)(

bbbe ++=∞

θθ (6.19)

where 3

)3(0)3(

02

LBR

λπ

θ = . (6.20)

Equation 6.17 and 6.19 indicate that as long as the steady-state phase difference is

available, the incoming signal acceleration can be obtained for a second order loop and

the incoming signal jerk can be obtained for a third order loop.

For an ARCTAN discriminator, theoretically its output is the phase difference if the PLL

is in a locked status. However, in reality the output of the discriminator consists of phase

79

differences and noise. The noise is so large that only noise can be seen if one looks at

the signal waveform from the discriminator. The next step is to extract the steady-state

phase difference from the noise.

In this research, two steps are taken to extract less noisy information. The first step is to

add the output of the discriminator together at every epoch and then multiply this sum

with the pre-integration time. Using Equation 6.17 and 6.18, for a second order loop, this

process is written as

)()(

2)()(

2)(2

101

)2(02

101kVel

BbbTnR

BbbT

L

k

nL

k

nne λ

πλ

πθ+

=+

=∞ ∑∑==

(6.21)

where k is the number of the pre-integration and )(kVel is the signal velocity.

Similarly, using Equation 6.19 and 6.20, for a third order loop, this process is written as

)()(

2)()(

2)(3

3101

)3(03

3101kAcce

BbbbTnR

BbbbT

L

k

nL

k

nne λ

πλ

πθ++

=++

=∞ ∑∑==

(6.22)

where )(kAcce is the signal acceleration.

The addition works as averaging and therefore the noise is reduced and the less noisy

measurements )(kVel and )(kAcce are obtained respectively for the second and third

order loops.

The second step is to calculate the approximate derivatives of )(kVel or )(kAcce .

Through this step, the signal acceleration is obtained for a second order loop and the

80

signal jerk is obtained for a third order loop. For the second order loop, this process is

written as

iTikVelkVelkAcce )()()( −−≈ (6.23)

where i is a positive integer.

Similarly, for the third order loop this process is written as

iTikAccekAccekJerk )()()( −−= (6.24)

where )(kJerk is the signal jerk.

Actually i determines the time interval in computing the approximated derivatives. It

should be noted that considering the noise influence, the time interval should not be too

small.

After the estimation of the signal dynamics, next step is to determine the proper

bandwidth.

6.3 Optimal bandwidth

As discussed before, the bandwidth selected is often a compromise design in order to

reduce different factors effects on the tracking performance. Here, we focus on the

thermal noise and dynamic stress influence and try to find an optimal bandwidth which

can decrease both errors at the same time. Considering the Equation 2.56, 2.57, 2.58 and

2.63 together, the following equation should be satisfied for the second order PLL:

81

o153

2809.0/2

11/2

3602

2

2

00≤+

+

n

n

BdtdR

NTCNCB

π (6.25)

The corresponding equation for the third order PLL is

o153

4828.0/2

11/2

3603

3

3

00≤+

+

n

n

BdtdR

NTCNCB

π (6.26)

Considering the effects of vibration induced phase noise and Allan deviation induced

phase noise, the threshold should be smaller than 15°.

In the following, Equation 6.26 is taken as an example for the explanation on how to

determine the optimal bandwidth using an iteration method. Here a 10° threshold is

assumed.

Step1: assume that the error is caused only by dynamic stress and calculate the required

bandwidth using the estimated jerk from Equation 6.24. The formula is written as

3 3

3

304828.0 dt

dR

Bn = (6.27)

Step 2: use the calculated bandwidth and carrier over noise power ratio acquired from the

software receiver (otherwise it has to be calculated) to calculate the error tσ caused by

thermal noise using Equation 2.56

Step 3: use the thermal noise error calculated in step 2 and threshold to calculate the

dynamic stress error eθ using the following equation

82

)10(3 te σθ −×= o

(6.31)

Step 4: use the dynamic stress error calculated in step 3 to calculate the required

bandwidth using

3 3

3

4828.0e

ndtdR

Bθ

= (6.32)

Step 5: iterate step 2, 3 and 4 until the bandwidth variation between two iterations is less

that a certain small value, 0.1Hz is chosen in this thesis.

83

CHAPTER SEVEN: TEST RESULTS AND ANALYSIS

The algorithms discussed in this thesis were tested using signals simulated with a GPS

hardware receiver, a GPS receiver�s front end which outputs a digitalized intermediate

frequency (IF) and the GNSS_SoftRx� software receiver developed by the PLAN

group. The first section introduces the configuration of the test including hardware and

software. In order to test the proposed algorithms performance, two tests are designed to

demonstrate the tracking performance under high dynamics applications � a horizontal

motion and a three-dimensional motion. The second and third section show test results

for real-time and post-processing scenarios. The statistical results from an ordinary PLL

and the proposed algorithms are presented in the last section.

7.1 Test Configuration

7.1.1 Test Scheme

The GSS 6560 GPS hardware simulator which can provide L1 C/A code signals under

various scenarios is used to generate the signals. An aircraft flight is chosen to generate

high dynamics motion. An Agilent Signal Generator (E 4431B) is used to generate white

noise which is input into a signal combiner GSS 4766 together with the output of the GPS

simulator. Therefore the output of the combiner provides simulated GPS signals with

noise. The generated white noise is utilized to change the Signal-to-Noise Ratio. A LNA

(low noise amplifier) is used to play the same role as the LNA in a real GPS antenna

84

which can provide a 30 dB gain. Connected with the LNA is a GPS front end made by

ACCORD Company, which can output digitized GPS IF signals with variable sample

frequencies ranging from 2 MHz to 20 MHz. At last, the software receiver does the signal

processing after IF and the computation of the navigation solution. The test configuration

is shown in Figure 7.1.

Figure 7.1: Test Configuration

Signal Tap

Data collection Software Receiver

LNA

SimGen Software

Agilent Signal Generator

GPS Simulator STR6550

Interference Combiner GSS4766

85

Two kinds of high dynamics are simulated. One is a horizontal motion and the other

one is a three dimensional motion. For the first two algorithms, three scenarios are tested

based on different noise levels and loop bandwidths. The signal power is configured at -

130 dBm to simulate an outdoor environment. The added white noise characteristics are

shown in table 7.1

Table 7. 1: Noise Characteristics

Central Frequency 1575.42 MHz

Bandwidth 2 MHz

Power1 -120 dBm

Power2 -90 dBm

The simulated data is 100 seconds long (limited by the Signal Tap). At the beginning, the

vehicle stays still for 30 seconds (which makes the acquisition easier) and then takes a

low acceleration for 10 seconds. Following that, a high dynamics motion is simulated.

7.1.2 GPS Receiver Configuration

The front end is the GPS Signal Tap, whose block diagram is shown in Figure 7.2. The

input GPS signal is amplified, filtered, down-converted, sampled and stored in a flash

memory. The flash memory can store 100 seconds of data. The main configuration of the

front end (Signal Tap) configuration is given in Table 7.2.

86

Figure 7.2: The Block Diagram of GPS Front End

Table 7. 2: Signal Tap Configuration

Frequency GPS L1 at 1575.42 MHz

GPS code C/A

Sampling Rate 4.75 MHz

IF bandwidth 2 MHz

Signal IF frequency 1.17 MHz

Quantization level 1 bit

The signal processing, which involves the signal acquisition, signal tracking and the

computation of the navigation solution is carried out in the software receiver. As

87

mentioned before, this research focuses on the PLL tracking performance. The PLL

design scheme and parameters are shown in table 7.3.

Table7. 3: PLL Configuration

Order of the PLL Three

Discriminator ARCTAN

PLL Bandwidth 18 Hz, 30 Hz

Pre-integration time 1 ms

7.2 Horizontal Motion Testing, Results and Analysis

The first test is a horizontal motion with 180° turns and lateral acceleration from 3.6 g to

4.6 g. The simulated vehicle trajectory produced by the simulator is shown in Figure 7.3.

The simulator does not provide the Doppler frequency directly. However, it supplies the

pseudorange rate. From this information, it is very easy to derive the Doppler frequency

using Equation 6.4. The derived Doppler frequency for satellite 9 is shown in Figure 7.4

as an example.

In addition, the acceleration and jerk can be derived by making use of the pseudorange

rate according to the following relations:

tv

dtdva

∆∆≈= (7.1)

88

ta

dtda

∆∆≈=α (7.2)

where a is the acceleration along the line of sight between the satellite and the user, α is

the jerk acceleration along the line of sight between the satellite and the user, t∆ is the

time interval.

Figure 7.3 Simulated Horizontal Vehicle Trajectory

10 20 30 40 50 60 70 80 90 400

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950True Doppler Frequency - Sat.9

Time (Sec)

Doppler Frequency (Hz)

Figure 7.4 Doppler Frequency of Satellite 9

-5 0 0 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 00

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

9 0 0

E a s t (m )

Nor

th (m

)

U s e r T ra je c to ry

3 . 6 g

4 . 4 g4 .0 g3 .6 g

4 . 6 g4 .2 g3 . 8 g

89

It is obvious that a smaller t∆ produces better accuracy. However, the data interval that

the simulator gives is 10 milliseconds which is not as small as we require. In the test, the

pre-integration time is configured to 1 ms therefore we can obtain the Doppler

frequencies output at 1-ms intervals. The Doppler frequency, acceleration and jerk from

the simulator output shown herein are not as accurate as desired for a 1 ms interval.

However they provide a reference to compare the tracking performance obtained from the

software receiver. Figure 7.5, 7.6 and 7.7 show the Doppler frequencies from an ordinary

PLL for satellite 9. For the first scenario, the bandwidth of the PLL is 18 Hz and the

C/NO is 45 dB-Hz. For the second scenario, the bandwidth is 18 Hz and the C/NO, 39 dB-

Hz. For the third scenario, the bandwidth is 30 Hz and C/NO, 39 dB-Hz.

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950Doppler Frequency From PLL - Sat.9

Time (Sec)

Doppler frequency (Hz)

Figure 7.5: Doppler frequency from an ordinary PLL for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

90

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Figure 7.5, 7.6 and 7.7 show that Doppler frequencies from the ordinary PLL get noisier

with the increase of the noise power and the bandwidth of the PLL. Further studying

91

Figure 7.5 and Figure 7.6 and 7.7, it is found that there are Doppler frequency offsets

between them. This is caused by the receiver clock�s frequency drift. The data used for

Figure 7.6 and 7.7 is the same but the data used for Figure 7.5 was collected at a different

time. Due to the random frequency drift of the receiver clock, the resulting Doppler

frequencies offsets randomly.

7.2.1 Real Time Processing

Real time processing herein means that the results were obtained using previous and

present data.

7.2.1.1 Results of Kalman Filter-Based Tracking Algorithm

Three scenarios are used in this test. The Doppler frequency obtained from the Kalman

filter-based tracking algorithm for three scenarios are shown in Figure 7.8, 7.9, 7.10.

Examining these figures and comparing them with Figure 7.5, 7.6 and 7.7, it is obvious

that the results from the Kalman filter-based tracking algorithm are less noisy than the

ordinary PLL results. A further study of these figures also leads to the conclusion that,

the lower signal-to-noise ratio, the better the tracking performance with the Kalman

filter-based algorithm. The reason is that when the measurements are noisy, the Kalman

filter-based algorithm puts more weight on its state model than on the measurement. As a

result, the estimated Doppler frequency provides better performance than the one from

the NCO.

92

However, if the Kalman filter-based tracking algorithm is used entirely instead of the

ordinary PLL for high dynamics applications, especially when the measurements are

noisy, the Kalman filter-based tracking algorithm will often diverge, especially for a fast

changing trajectory. The problem is caused by the fast and random change in the phase

difference, Doppler frequency and change rate of the Doppler frequency during the

updating interval which cannot be modelled correctly. Consequently the estimations of

these parameters are inaccurate. In this case, an inaccurate Doppler frequency will be

used for Doppler frequency removal. Therefore the phase difference measurement from

the output of the discriminator can not provide useful information. As a result the

divergence occurs. To deal with this problem, this research combined the ordinary PLL

with the Kalman filter-based tracking algorithm to provide more reliable tracking. The

PLL can guarantee the reliable phase lock even in high dynamic situations if an

appropriate bandwidth is chosen. Even though the estimations of Kalman filter are not

accurate due to the inaccurate system model, the Kalman filter can correct its estimations

using the accurate phase difference measurement. Therefore a reliable tracking is

guaranteed.

93

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950Estimation of Doppler Frequency - Sat.9

Time (Sec)


Figure 7.8: Doppler frequency from Kalman filter-based tracking algorithm for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

10 20 30 40 50 60 70 80 90400

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Time (Sec)



94

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Time (Sec)



Figure 7.11, 7.12, 7.13 indicate the PLL Lock status. The Y axis of these figures

represents the cosine values of the phase differences. When the PLL is locked, the phase

difference should be zero therefore its cosine value should be 1. However due to the noise

effect, it is not possible to obtain exactly 1. It is however safe to assume that, if the cosine

value is greater than 0.8, the PLL is locked. When carefully examining the output from

the PLL Lock Indicator, it is found that sometimes the PLL lost lock. A further

investigation reveals that the phase status becomes worse as the noise power increases.

However the phase status improves with an increase in bandwidth. The former result is

easier to understand. For a given bandwidth, the PLL tracking error increases with the

increase of the noise. When the tracking error is larger than the lock threshold, the PLL

loses lock. The latter result can be explained in this way: under the high dynamic

situations, Doppler frequency changes fast with time that requires PLL adjusts its

95

frequency quickly to catch this change trend. As long as the frequency difference

between the incoming signal and the NCO is within the Lock range, the PLL can still

lock within one single-beat note between the incoming signal and the NCO frequencies.

The lock time is inversely proportional to the PLL bandwidth. With the help of the wider

bandwidth, more control signals pass through the loop filter and affect the NCO.

Consequently, the NCO adjusts itself much faster to tracking the changes in the incoming

signal.

10 20 30 40 50 60 70 80 900

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0.7

0.8

0.9

1PLL Lock Indicator - Sat.9

Time (Sec)

Cos( phase difference)

Figure 7.11: PLL lock indicator for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

96

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0.9


Time (Sec)


Figure 7.12: The PLL lock indicator for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz)

10 20 30 40 50 60 70 80 900

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Time (Sec)



97

7.2.1.2 Results Using the Wavelet De-noising Technique

The wavelet db16 (Daubechies wavelet) and 10 levels decomposition are chosen herein.

Figures 7.14, 7.15 and 7.16 illustrate the results from the wavelet de-noising technique.

These figures demonstrate improvements compared with Figures 7.5, 7.6 and 7.7. Even

though the improvements are significant with the increase of the noise power and

bandwidth, the results are poorer when compared with those from the Kalman filter-

based tracking algorithm (Figures 7.8, 7.9 and 7.10). It should be noted that the wavelet

de-noising technique can only mitigate the noise level but cannot eliminate the noise

totally. The remaining noise which is within the bandwidth can still pass through the loop

filter and affect the NCO tracking performance.

10 20 30 40 50 60 70 80 90400

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950Doppler Frequency after Wavelet De-noising - Sat.9

Time (Sec)


Figure 7.14: Doppler frequency after applying wavelet de-noising technique in the PLL for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

98

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99

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Time (Sec)

Cos(phase difference)


100

10 20 30 40 50 60 70 80 90 0

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0.8

0.9

1PLL Lock Indicator- Sat.9

Time (Sec)

Cos( Phase Difference)

Figure 7.19: The PLL lock indicator for satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) Similar to the Kalman filter-based tracking algorithm results, Figure 7.17, 7.18 and 7.19

indicate that the PLL loses lock more frequently with the decrease of the signal-to-noise

ratio. The wider bandwidth keeps the PLL locked.

The de-noising performance is related to the selection of the wavelet and decomposition

levels. In theory, the decomposition can proceed until the individual detail consists of a

single sample. In practice, you will select a suitable number of levels based on the nature

of the signal or a suitable criterion such as entropy (The Mathwork, Inc. 2004). In

general, the higher the decomposition level, the better the de-noising performance will

be. However, after a certain level, the improvement is not very significant. Particularly

for real time applications, the higher decomposition level will take longer time to process.

101

Considering the above factors, 10 decomposition levels are chosen in this thesis. The

test results are shown below.

7.2.1.3 Results from Adaptive Bandwidth Algorithm

As discussed in Chapter 6, in order to adjust the PLL bandwidth, the first task is to

estimate the dynamics of the incoming signal. Figure 7.20 and 7.22 show the signal

acceleration and jerk derived from the pseudorange rate given by the simulator. Figure

7.21 and 7.23 show the estimated acceleration and jerk using the adaptive bandwidth

algorithm. These figures indicate that the adaptive bandwidth algorithm can estimate the

incoming signals dynamics. However, the estimated acceleration and jerk are noisy.

Ignoring the clock errors (caused by vibration and Allan deviation), the PLL bandwidth is

adapted according to the acceleration (second order loop) and jerk (third order loop) and

signal-to-noise ratio. The noisy dynamics estimation decreases the accuracy of the

adaptive bandwidth.

102

10 20 30 40 50 60 70 80 90

-100

-80

-60

-40

-20

0

20

40

60

80

100

True Acceleration - Sat.9

Time (Sec )

Acceleration (m/s

2)

Figure 7.20: True acceleration from the simulator for satellite 9

10 20 30 40 50 60 70 80 90

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-80

-60

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-20

0

20

40

60

80

100

Estimated Acceleration - Sat.9

Time (Sec)

Acceleration (m/s

2)

Figure 7.21: Estimated acceleration using the adaptive bandwidth algorithm for satellite 9

103

10 20 30 40 50 60 70 80 90 -60

-40

-20

0

20

40

60True Jerk - Sat.9

Time (Sec)

Jerk (m/s

3)

Figure 7.22: True jerk from the simulator for satellite 9

10 20 30 40 50 60 70 80 90-60

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-20

0

20

40

60Estimated Jerk - Sat.9

Time (Sec)

Jerk (m/s

3)

Figure 7.23: Estimated jerk using the adaptive bandwidth algorithm for satellite 9

Figure 7.24 shows the Doppler frequency after adapting the bandwidth for satellite 9 and

when the C/NO is 45 dB-Hz. A comparison of Figure 7.24 and 7.5 shows the

104

improvement. Figure 7.25 shows the adaptive bandwidth which changes with the

estimated dynamics. Figure 7.26 shows the PLL lock indicator results. The figure

demonstrates that sometimes the PLL still loses lock even though the bandwidth is

adaptable. There are two factors that affect this phenomenon. First when computing the

adaptive bandwidth, the clock effects are not taken into account. The resulting bandwidth

is not as accurate as it should be. Secondly under high dynamics situations, the bandwidth

must change fast to catch up with the fast variations in the incoming signal. However the

PLL does not allow its bandwidth to change quickly, otherwise the loop filter will output

impulses which will cause the PLL to lose lock (Legrand & Macabiau 2000).

A simple method to deal with this problem is to insert a low-pass filter which will delay

the change rate of the PLL bandwidth. The response time of the low-pass filter is critical

for the abrupt variation in an incoming signal�s jerk which requires fast response in the

bandwidth. However, too short a response time can cause the loop to lose lock. This is a

major problem for the adaptive bandwidth algorithm.

105

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950Doppler Frequency after Adapting the Bandwidth - Sat.9

Time (Sec)


Figure 7.24: Doppler frequency after adapting the bandwidth for satellite 9 (C/NO is 45 dB-Hz)

10 20 30 40 50 60 70 80 90 10

11

12

13

14

15

16

17

18Adaptive Bandwidth - Sat.9

Time (Sec)

Bandwidth (Hz)

Figure 7.25: Adaptive bandwidth for satellite 9

106

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1PLL Lock Indicator- Sat.9

Time (Sec)

Cos(Phase Difference)

Figure 7.26: PLL lock indicator for satellite 9

7.2.2 Post Processing

In this thesis, all results from the three algorithms were processed again using the wavelet

de-noising technique. For this mode, all data involves in the de-noising processing. This

means except the last moment result, all other moment results are derived using the

previous, present and future information. The results indicate that better results are

obtained in post processing compared to real time processing.

7.2.2.1 Results from the Kalman filter-based tracking algorithm

The post processed results for the Kalman filter-based tracking algorithm are shown in

Figure 7.27, 7.28 and 7.29. The results clearly indicate an improvement, particularly for

the high noise level and wide bandwidth case.

107

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950Postmission Estimation of Doppler Frequency Using Kalman Filter - Sat.9

Time (Sec)


Figure 7.27: Doppler frequency from Kalman filter based tracking algorithm for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

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108

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Time (Sec)


Figure 7.29: Doppler frequency from Kalman filter-based tracking algorithm for satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) 7.2.2.2 Results from applying the wavelet de-noising technique in PLL

The post processed results, after applying the wavelet de-noising technique, are shown in

Figure 7.30, 7.31 and 7.32. The results show that the improvement is not as obvious as

with the Kalman filter. However compared with the real time results reported earlier in

Figure 7.14, 7.15 and 7.16, these post-processed results do provide some improvements.

109

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950Postmission Doppler Frequency After Wavelet De-noising - Sat.9

Time (Sec)


Figure 7.30: Doppler frequency after applying wavelet de-noising technique for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

10 20 30 40 50 60 70 80 90 400

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110

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Time (Sec)



7.2.2.3 Results from Adaptive Bandwidth Algorithm

The post processed result for the adaptive bandwidth algorithm is shown in Figure 7.33.

Compared to the real time result of Figure 7.24, the improvement is significant.

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950Postmission Doppler Frequency after Adapting the Bandwidth - Sat.9

Time (Sec)


111

Figure 7.33: Doppler frequency after adapting the PLL bandwidth for satellite 9 (C/NO is 45 dB-Hz)

7.3 3-Dimensional Motion Test, Results and Analysis

In this test, a three dimensional motion with a 360° turn and a 3.8 g lateral acceleration in

the horizontal plane and simultaneously, 5 m/s velocity in the height direction, is

simulated. The horizontal and 3-D trajectories are shown in Figure 7.34 and 7.35.

Figure 7.34: 3-Dimensional trajectory 360º turns and 3.8 g lateral acceleration in horizontal plane and 5 m/s velocity in height direction.

-5000

5001000

1500

0

1000

2000

3000-100

0

100

200

300

East (m)

Three Dimensional Trajectory

North (m)

Up (m)

112

Figure 7.35: Horizontal trajectory with 360º turns and 3.8 g lateral acceleration

Similarly, the Doppler frequency can be derived from the pseudorange rate given by the

simulator. That for satellite 9 is shown in Figure 7.36.

0 10 20 30 40 50 60 70 80 90 100

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1100True Doppler Frequency - Sat.9

Time (Sec)


Figure 7.36: Doppler frequency from the simulator for satellite 9

-2 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 00

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

E a st (m )

Nor

th (m

)

H o r i z o n ta l T r a j e c to ry

113

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1100Doppler Frequency from PLL - Sat.9

Time (Sec)



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1000


Time (Sec)

Doppler frequency Hz

Figure 7.38: Doppler frequency from an ordinary the PLL for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz)

114

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1100Doppler Frequency from PLL - Sat.9

Time (Sec)


Figure 7.39: Doppler frequency from an ordinary the PLL for satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz)

Again, the test is carried out for three scenarios. The Doppler frequencies from the

ordinary PLL for these three scenarios are shown in Figure 7.37, 7.38 and 7.39. Similar to

the test 1, Doppler frequencies become very noisy with the increase of the noise power

and the loop bandwidth.

7.3.1 Real Time Processing

7.3.1.1 Results from the Kalman Filter-Based Tracking Algorithm

Figure 7.40, 7.41 and 7.42 show the Doppler frequencies from the Kalman filtered-based

tracking algorithm and Figure 7.43, 7.44 and 7.45, their corresponding PLL lock

115

indicators. The maximum jerk in this test is higher than the one in the horizontal

motion. However, due to the PLL existing, the Doppler frequency from the Kalman filter-

based tracking algorithm can still track the input Doppler frequency correctly. Even

though the PLL sometimes lose phase lock, as long as it keeps frequency tracking, the

Kalman filter-based tracking algorithm can still obtain the correct phase different

measurements from the output of the PLL discriminator. This ensures accurate Doppler

frequency measurements, a process which is essential for carrier phase measurements.

The same phenomena exist for this test. As the noise power increases, the Doppler

tracking performance does not deteriorate very much. This also applies to the scenario of

a 30 Hz PLL bandwidth. Obviously the Doppler frequency from the ordinary PLL is

much noisier due to a wider bandwidth. However, the Kalman filter-based tracking

algorithm generates almost the same result as for the 18 Hz PLL bandwidth case. This

leads to the conclusion that the Kalman filter-based tracking algorithm still behaves well

in low signal-to-noise ratio and wider bandwidth situations.

116

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1000


Time (Sec)


Figure 7.40: Doppler frequency from the Kalman filter-based tracking algorithm for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

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Time (Sec)

Doppler frequency Hz


117

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7.3.1.2 Results with the Wavelet De-noising Technique

119

Figures 7.46, 7.47 and 7.48 show the Doppler frequencies when applying wavelet de-

noising technique in the PLL and Figures 7.49, 7.50 and 7.51 their corresponding PLL

lock indicators. Again, these figures show that the PLL tracking performance improve

significantly with an increase of the noise power and bandwidth and the PLL more easily

loses phase lock when the noise power increases and the bandwidth narrows.

10 20 30 40 50 60 70 80 90

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Time (Sec)


Figure 7.46: Doppler frequency after applying wavelet de-noising technique in PLL for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

120

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1PLL Lock Detector - Sat.9

Time (Sec)



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1PLL Lock Detector - Sat.9

Time (Sec)



7.3.1.3 Results from the Adaptive bandwidth algorithm

Figures 7.52 and 7.54 show the true acceleration and jerk derived from the pseudorange

rate given by the simulator and Figures 7.53 and 7.55 show the estimated acceleration

and jerk using the adaptive bandwidth algorithm. Obviously the estimated acceleration

and jerk can follow the changes in the real acceleration and jerk quite well. The biggest

problem is that both estimations are noisy - especially the jerk, which is derived from

acceleration. Even though the jerk in this test is not, on the average, higher than the jerk

in test 1, the rate change of the jerk in test 2 is sometimes much higher than the one in

test 1. Therefore it is more difficult for the adaptive bandwidth to keep lock.

123

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100True Acceleration - Sat.9

Time (Sec)

Acceleration (m/s

2)

Figure 7.52: The true acceleration from the simulator for satellite 9

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100Estimated Acceleration - Sat.9

Time (Sec)

Acceleration (m/s

2)

Figure 7.53: The estimated acceleration using the adaptive bandwidth algorithm for satellite 9

124

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40True Jerk - Sat.9

Time (Sec)

Jerk (m/s

3)

Figure 7.54: The true jerk from the simulator for satellite 9

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0

20

40Estimated Jerk - Sat.9

Time (Sec)

Jerk (m/s

3)

Figure 7.55: The estimated acceleration using the adaptive bandwidth algorithm for satellite 9

125

Figure 7.56 and 7.57 show the Doppler frequency after adapting the bandwidth.

Figure 7.57 indicates that the PLL bandwidth changes with the incoming dynamics.

10 20 30 40 50 60 70 80 90

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1100Doppler Frequency after Adapting the Bandwidth - Sat.9

Time (Sec)


Figure 7.56: The Doppler frequency after adapting the bandwidth for satellite 9 (C/NO is 45 dB-Hz)

10 20 30 40 50 60 70 80 90 10

11

12

13

14

15

16

17

18Adaptive Bandwidth - Sat.9

Time (Sec)

Bandwidth (Hz)

Figure 7.57: Adaptive bandwidth for satellite 9

126

7.3.2 Post Processing

7.3.2.1 Results from the Kalman Filter-Based Tracking

The post processed results for the Kalman filter-based tracking algorithm are shown in

Figure 7.58, 7.59 and 7.60. The results clearly indicate improvement particularly for a

high noise level and wide bandwidth case.

10 20 30 40 50 60 70 80 90

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1000


Time (Sec)


Figure7. 58: Doppler frequency from Kalman filter based tracking loop for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 45 dB-Hz)

127

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Time (Sec)


Figure7. 59: Doppler frequency from Kalman filter based tracking loop for satellite 9 (PLL Bandwidth = 18 Hz, C/NO = 39 dB-Hz)

10 20 30 40 50 60 70 80 90

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1000


Time (Sec)


Figure 7.60: Doppler frequency from Kalman filter based tracking loop for satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz)

128

7.3.2.2 Results from Applying the Wavelet De-noising Technique

The post processed results for applying the wavelet de-noising technique are shown in

Figure 7.61, 7.62 and 7.63. The results show improvement particularly for the high noise

power and wider bandwidth compared to Figure 7.46, 7.47 and 7.48.

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1000


Time (Sec)


Figure 7.63: Doppler frequency after applying wavelet de-noising technique for satellite 9 (PLL Bandwidth = 30 Hz, C/NO = 39 dB-Hz) 7.3.2.3 Results from the Adaptive Bandwidth Algorithm

The post processed result for the adaptive bandwidth algorithm is shown in Figure 7.64.

Compared to the real time result of Figure 7.56, the improvement is significant.

130

10 20 30 40 50 60 70 80 90

300

400

500

600

700

800

900

1000

1100Postmission Doppler Frequency after Adapting the Bandwidth - Sat.9

Time (Sec)


Figure 7.64: Doppler frequency from adaptive bandwidth algorithm for satellite 9 (C/NO is 45 dB-Hz)

7.4 Summary

This section presents statistic results for the two tests and scenarios. Table 7.4 and 7.5

shows the real-time results for the horizontal motion and the three dimensional motion,

respectively. The results for an ordinary PLL and the algorithms proposed in this thesis

for three scenarios are presented. Both tables show a similar trend for the two tests. The

ordinary PLL produces the highest maximum and minimum Doppler frequency error.

The Kalman filter-based tracking algorithm demonstrates the best performance for the

above two cases. The wavelet de-noising technique provides medium improvement,

while the adaptive bandwidth algorithm provides little improvement. For both tests, the

Doppler frequency error standard deviation for an ordinary PLL and the proposed

algorithms deteriorates with an increase in the noise and PLL bandwidth. Again, the

131

Kalman filter-based tracking algorithm shows the best improvement, which becomes

significant with an increase in the noise and PLL bandwidth. This result matches well

those shown in Figure 7.8, 7.9 and 7.10. The wavelet de-noising algorithm also shows

improvement, which are however less than those obtained with the Kalman filter-based

tracking algorithm, but have similar trends.

Table 7. 4: Real-Time Statistical Results for the Horizontal Motion Doppler Frequency Error Ordinary PLL

(Hz) Kalman Filter (Hz)

Wavelet De-noising

(Hz)

Adaptive Bandwidth

(Hz) Max

9.7 4.4 6.4 9.2

Min

-9.8 -5.1 -7.2 -8.7

Mean

-0.1 0.0 0.1 0.0

Bn=18 Hz

C/NO=45

dB-Hz Std Dev

2.6 1.7 1.9 2.4

Max

11.7 5.5 7.2

Min

-11.9 -5.1 -8.3

Mean

-0.1 0.0 -0.1

Bn=18 Hz

C/NO=39

dB-Hz Std Dev

3 1.7 2.0

Max

18.5 5.6 8.0

Min

-17.7 -6.0 -8.2

Mean

0.0 -0.1 0.3

Horizontal

Motion

Bn=30 Hz

C/NO=39

dB-Hz Std Dev

4.0 1.7 2.3

132

Table 7. 5: Real-Time Statistic Results for the Three Dimensional Motion

Doppler Frequency Error Ordinary PLL (Hz)

Kalman Filter (Hz)

Wavelet De-noising

(Hz)

Adaptive Bandwidth

(Hz) Max

9.3 5.0 7.1 8.0

Min

-8.3 -4.7 -5.4 -7.3

Mean 0.0 0.0 -0.1 -0.1

Bn=18 Hz

C/NO=45

dB-Hz Std Dev

2.3 1.5 1.8 2.1

Max

14.3 6.5 7.5

Min

-13.2 -6.2 -7.3

Mean 0.1 0.0 0.0

Bn=18 Hz

C/NO=39

dB-Hz Std Dev

3.4 2.1 2.4

Max

17.8 6.5 8.0

Min

-18.1 -6.0 -8.2

Mean 0.0 0.0 -0.2

Three

Dimensional

Motion

Bn=30 Hz

C/NO=39

dB-Hz Std Dev

4.4 2.1 2.4

Post processed results are shown in Table 7.6 and 7.7. It is obvious that a much better

performance is obtained, compared to the real-time results. The Kalman filter-based

tracking algorithm still demonstrates the best performance. However, the wavelet-de-

noising algorithm does not provide a large improvement compared to the other two

algorithms. This indicates that if the wavelet de-noising technique is applied to a signal

which is already processed by the wavelet de-noising technique in real-time, not too

much improvement can be expected.

133

Table 7. 6 : Post Processed Statistic Results for the Horizontal Motion

Doppler Frequency Error Kalman Filter (Hz)

Wavelet De-noising

(Hz)

Adaptive Bandwidth

(Hz) Max

2.5 4.2 3.8

Min

-3.3 -4.5 -4.2

Mean

-0.1 0.0 0.0

Bn=18 Hz

C/NO=45

dB-Hz Std Dev

1.2 1.6 1.3

Max

3.4 4.7

Min

-4.1 -4.7

Mean

0.0 0.0

Bn=18 Hz

C/NO=39

dB-Hz Std Dev

1.2 1.8

Max

3.8 5.2

Min

-4.0 -4.8

Mean

0.0 0.2

Horizontal

Motion

Bn=30 Hz

C/NO=39

dB-Hz Std Dev

1.2 1.8

134

Table 7. 7: Post Processed Statistic Results for the Three Dimensional Motion

Doppler Frequency Error Kalman Filter (Hz)

Wavelet De-noising

(Hz)

Adaptive Bandwidth

(Hz) Max

3.0 4.3 3.2

Min

-2.8 -4.1 -4.0

Mean

0.0 -0.1 -0.1

Bn=18 Hz

C/NO=45

dB-Hz Std Dev

1.0 1.6 1.3

Max

4.1 4.5

Min

-3.5 -5.3

Mean

0.0 0.0

Bn=30 Hz

C/NO=39

dB-Hz Std Dev

1.4 2.0

Max

4.0 4.1

Min

-4.4 -4.6

Mean

-0.1 0.0

Three

Dimensional

Motion

Bn=30 Hz

C/NO=39

dB-Hz Std Dev

1.4 2.1

Table 7.8 and 7.9 present the Doppler frequency improvement obtained from the two

tests and different scenarios. In general, all results comply with the results from previous

figures. Better performances are obtained from the Kalman filter-based tracking

algorithm and the wavelet de-noising algorithm. However the adaptive bandwidth

algorithm yields a poor improvement particularly for the real-time processing. This is

because the minimum PLL bandwidth is set to 10 Hz which is not narrow enough in

order to obtain a better improvement. The reason for using a 10 Hz minimum PLL

135

bandwidth is to guarantee PLL tracking reliability. As discussed before, a PLL does

not allow its bandwidth to be changed too fast otherwise it will lose lock. However, a

high dynamic signal requires a rapid variation of the PLL bandwidth, and a 10 Hz PLL

bandwidth can prevent this from happening.

Table 7. 8: Real-Time Doppler Frequency Improvement

Doppler Frequency Improvement

Scenarios

Kalman

Filter

Wavelet

De-noising

Adaptive

Bandwidth

Bn=18 Hz C/NO=45

dB-Hz

35%

26%

7%

Bn=18 Hz C/NO=39

dB-Hz

41%

30%

Horizontal

Motion

Bn=30 Hz C/NO=39

dB-Hz

56%

43%

Bn=18 Hz C/NO=45

dB-Hz

34%

23%

10%

Bn=18 Hz C/NO=39

dB-Hz

39%

30%

Three

Dimensional

Motion Bn=30 Hz C/NO=39

dB-Hz

52%

44%

136

Table 7. 9: Post-Processing Doppler Frequency Improvement

Doppler Frequency Improvement

Scenarios

Kalman

Filter

Wavelet

De-noising

Adaptive

Bandwidth

Bn=18 Hz C/NO=45

dB-Hz

55%

39%

50%

Bn=18 Hz C/NO=39

dB-Hz

58%

43%

Horizontal

Motion

Bn=30 Hz C/NO=39

dB-Hz

69%

56%

Bn=18 Hz C/NO=45

dB-Hz

55%

32%

42%

Bn=18 Hz C/NO=39

dB-Hz

59% 40%

Three

Dimensional

Motion Bn=30 Hz C/NO=39

dB-Hz

68%

52%

137

CHAPTER EIGHT: CONCLUSIONS AND RECOMMENDATIONS

Three algorithms to improve the tracking performance of the PLL are presented in

Chapter 4, 5 and 6. These algorithms were developed and implemented as a part of a

GPS software receiver. Then they were tested and compared to the results using an

conventional GPS software receiver.

8.1 Conclusions

The Kalman filter-based tracking algorithm uses the tracking ability of an ordinary PLL

as a basis and augments it with a carrier phase dynamics model to estimate the phase

difference, the Doppler frequency and its rate change. The purpose of keeping the

ordinary PLL is to guarantee a good tracking reliability. Even though the output of the

PLL discriminator is very noisy, the carrier phase dynamic model can still extract

valuable information from it. By varying the weight on the measurement according to its

noise power level, the Kalman filter-based tracking algorithm demonstrates a significant

improvement. The estimated Doppler frequency and phase difference can be used as

input to derive the carrier phase measurement. The test results shown in Chapter 7

demonstrate that this algorithm works especially well under low signal-to-noise ratios

and high dynamic applications that require wide bandwidths.

Applying a wavelet de-noising technique in the PLL helps reducing the noise within the

bandwidth of the loop filter and therefore, a less noisy tracking performance can be

obtained by the NCO. Even though the noise is reduced, the PLL is still not 100%

138

effective due to the effect of high dynamic stress. However this disadvantage can be

overcome by increasing the bandwidth. Results from Chapter 7 indicate that, under the

conditions of wide bandwidth and a low signal-to-noise ratio, this algorithm produces

superior tracking performance. Even though the improvement is not as significant as that

resulting from the Kalman filter-based tracking algorithm, applying the wavelet de-

noising technique in the PLL is still an effective method to improve the PLL tracking

performance, particularly in post processing mode.

The key premise of the successful use of the adaptive bandwidth algorithm is the correct

estimation of incoming signal dynamics. The adaptive bandwidth algorithm can estimate

the dynamics quite well under normal signal-to-noise ratio and high dynamics but with

relatively less random variation. As a result, the bandwidth can be adapted correctly

according to the dynamics and the tracking performance is improved specifically for the

period of low dynamics. However, if the signal-to-noise ratio is low or dynamics changes

are very fast in a short interval, some problems will arise. Because the measurements

used for the estimation are noisy, the estimation cannot avoid the impact of the noise and

the estimation accuracy will decrease. Consequently the bandwidth will be adapted in an

incorrect way. On the other hand, if dynamics changes very fast and randomly, the PLL

bandwidth has to be changed fast to follow this change of trend. However as discussed in

Chapter 6, the PLL does not allow its bandwidth to change fast otherwise lost of phase

lock will result. In the worst case, the PLL will not only lose phase lock but also will

lose frequency lock. These two problems restrict the application of the adaptive

bandwidth algorithm.

139

Post processing results show better improvements especially for the Kalman filter-

based tracking algorithm and adaptive bandwidth algorithm than the results for the real

time processing case. These improvements further show that the wavelet de-noising

technique can be applied to improve noisy measurements such as pseudoranges.

8.2 Recommendations

Based on the test results in the thesis, the following recommendations can be made: The

Kalman filter-based tracking algorithm provides better improvements than the other two

algorithms. The combination of the PLL and the carrier phase dynamic model improves

the tracking reliability particularly in the case of high and random dynamics applications.

Based on the theory, the measurement is not restricted at the output of the PLL

discriminator. Measurements from other points are possible such as the Doppler

frequency from the output of the NCO.

Applying the wavelet de-noising technique in the PLL is another promising method.

Actually this algorithm is not limited in improving the tracking performance applications

but also other applications that relate to noisy measurements such as pseudoranges. In

addition, the selection of the wavelet and decomposition levels also plays some role in

the final performance. Further, the combination of the wavelet de-noising technique and

the other two algorithms offers potential. In addition, the de-noising technique can also

be applied at the output of the loop filter.

140

Using an adaptive bandwidth algorithm must be carefully considered under a low

signal-to-noise ratio. If considering all tracking errors including clock errors caused by

vibration and Allan deviation, perhaps a more accurate bandwidth can be obtained.

The combination of the Kalman filter-based tracking algorithm with the wavelet de-

noising technique may provide further improvement. In this scheme, the wavelet de-

noising should be applied immediately after the PLL discriminator and the result after

wavelet de-noising used as the phase difference measurement for the carrier phase

dynamic model. The less noisy measurement is likely to provide a significant

improvement.

Another possible combination is between the adaptive bandwidth algorithm and the

wavelet de-noising technique. Again, the wavelet de-noising technique should be applied

immediately after the discriminator. In this way, a less noisy phase difference is provided

to the adaptive bandwidth algorithm. From previous discussion, it is known that the phase

difference is an important measurement for the adaptive bandwidth algorithm to estimate

the dynamics. Therefore combining these two algorithms together will improve the

performance of the adaptive bandwidth algorithm in the case of low signal-to-noise

power ratios

There are two tests that could be done to enhance the results presented in this thesis. In

all three algorithms, the clock influence is not taken into account. If its effect can be

removed, better tracking performance could be obtained. Particularly for the adaptive

bandwidth algorithm, a more accurate bandwidth estimation is possible.

141

Integrating GPS with other sensors is another area which has been studied in high

dynamic applications. The biggest challenge for this algorithm is the accuracy of the

aiding Doppler frequency which mainly depends on the quality of the sensors. The

accurate aiding Doppler can effectively narrow the bandwidth of the PLL and

consequently improve its tracking performance. In this case, the errors related to the

clock become the dominant error sources.

142

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146

Appendix A

DERIVATION OF THE STEADY-STATE ERROR

From the Section 6.2 we obtain

[ ]

+−

−−=−=∞ ∑∑

∞

=−

=

−−−

−

→→][

!)1(

)1()1(lim)()1(lim)(1

)(

1

0

11

1

11

0 m

m

m

N

n

nn

N

Nd

zeze kZmzbKz

zKzzzθ

θθ (A.1)

From the characteristic of the Z transform, we get

)]([)]([ nxZdzdznnxZ −= (A.2)

For )()( nunx = , Equation C.2 can be rewritten as

)]([)]([ nuZdzdznnuZ −= (A.3)

Because 1

)]([−

=z

znuZ

5

234

4

33

32

2

)1()2263()]([

)1(52)]([

)1()1()]([

)1()]([

−++−−=

−++−=

−−=

−=

zzzzznunZ

zzznunZ

zzznunZ

zznnuZ

(A.4)

147

+−

+−

+−

+−+−

−−=

+

−++−−

+

+−

++−+

−+

+−+−

−−=∞

∑

∑

−

=

−−−

−

→

−

=

−−−

−

→

])1()1()1()1(

[)1(

)1()1(

])1(24

)2263(

)1(6)52(

)1(2)1(

)1([

)1(

)1()1()(

5

)4(

4

)3(

3

)2(

2

)1(

1

0

11

1

1

5

)4(23

4

)3(3

3

)2(

2

)1(

1

0

11

1

1

0000

0

000

lim

lim

L

L

zzzzzbKz

zKz

zzzzz

zzz

zzz

zz

zbKz

zKz

N

n

nn

N

Nd

z

N

n

nn

N

Nd

ze

θθθθ

θ

θθθθ

(A.5)

From the above equation, it is obvious that for the items

>∞

=

<

=∞∑

−

=Nm

NmbK

Nm

N

nnO

N

e 1

0

0

0

)(θ

θ (A.6)

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Thesis Front Matter - University of Calgary